r 


REESE  LIBRARY 


-n_n__rv—rL--n_n, 


UNIVERSITY  OF  CALIFORNIA. 
A 

Deceived 
Accession  No.  /  6~  J  J  J      .   Class  No. 


MOLECULES 


MOLECULAR  THEORY 


MATTEE 


BY 


A.  D.   RISTEEN,   S.B. 


BOSTON,  U.S.A.,  AND  LONDON 

PUBLISHED     BY     GINN    &    COMPANY 

1896 


lirlll 

COPYRIGHT,  1895 
BY    A.    D.    KISTEEN 

ALL  RIGHTS  RESERVED 


PREFACE. 


IN  the  multiplication  of  popular  books  on  scientific  subjects, 
the  molecular  theory  of  matter  appears  to  have  been  strangely 
neglected.  None  of  the  works  available  to  American  readers 
pretend  to  give  a  complete,  connected  account  of  what  is 
known  of  the  constitution  of  matter,  and  the  student  who 
wishes  to  learn  the  present  state  of  the  molecular  theory  has 
to  seek  his  information  in  the  occasional  articles  that  are 
scattered  through  the  scientific  journals.  Dr.  Watson's  Kinetic 
Theory  of  Gases  (a  new  edition  of  which  has  been  recently 
published)  is  far  too  difficult  for  the  undergraduates  in  our 
scientific  schools  and  colleges ;  Clausius's  Kinetische  Theorie 
der  Gase  (1889-91)  has  not  yet  been  translated,  nor  has 
Meyer's  Kinetische  Theorie  der  Gase,  so  far  as  I  am  aware. 
Meyer's  book  is  also  out  of  print  at  present,  although  a  new 
edition  is  in  preparation.  Lord  Kelvin's  delightful  lecture 
on  The  Size  of  Atoms  should  be  read  by  all  students  of 
physics,  and  it  is  now  readily  available,  in  the  first  volume  of 
his  Popular  Lectures  and  Addresses.  Crookes's  classical  papers 
on  radiant  matter  should  also  be  read;  they  are  in  the 
Proceedings  of  the  Royal  Society,  beginning  with  the  year 
1874. 

The  present  volume  is  an  attempt  to  elucidate  the  elements 
of  the  molecular  theory  of  matter  as  it  is  held  to-day.  It  is 
based  on  a  lecture  delivered  on  the  12th  of  last  February, 
before  the  Washburn  Engineering  Society,  of  the  Worcester 
Polytechnic  Institute.  In  preparing  the  manuscript  for  the 
printer  a  considerable  number  of  alterations  have  been  made, 
and  much  new  material  has  been  added,  though  the  form  of 
presentation  has  been  preserved.  Special  care  has  been  takeni 


iv  PREFACE. 

to  exclude  all  matter  except  that  which  has  an  immediate  and 
evident  bearing  on  molecules;  otherwise  this  book  would  have 
been  a  treatise  on  physics.  For  this  reason  many  exceedingly 
interesting  theories  and  discoveries  have  been  omitted  or 
dismissed  with  slight  mention  —  such,  for  example,  as  the 
investigations  of  Ostwald  and  others  on  solutions.  It  could 
hardly  be  expected  that  different  men  would  agree  on  what 
should  be  passed  over  in  this  way,  and  it  is  likely  that  better 
judgment  might  have  been  "used  in  many  places.  It  is  also 
likely  that  there  have  been  many  oversights  and  accidental 
omissions.  In  some  cases  important  theories  have  been 
passed  over  because  it  was  believed  that  they  could  not  be 
adequately  discussed  without  introducing  considerable  digres- 
sions upon  the  phenomena  they  are  intended  to  explain. 
Professor  Ewing's  theory  of  magnetism  is  an  example  of  this, 
and  Clausius's  theory  of  electrolysis  has  been  dismissed  with 
a  bare  mention,  for  like  reasons. 

Throughout  this  volume  I  have  considered  molecules  in 
their  physical  aspect.  There  are  numerous  excellent  works 
that  discuss  the  chemical  aspect  of  the  molecular  theory 
satisfactorily,  of  which  the  following  may  be  particularly 
recommended:  Meyer's  Modern  Theories  of  Chemistry,  Hem- 
sen's  Theoretical  Chemistry,  Ostwald's  Outlines  of  General 
Chemistry,  and  Mendeleieff's  Principles  of  Chemistry. 

A.  D.  RISTEEN. 
HARTFORD,  CONN., 

September  1,  1894- 


.     GENERAL  CONSIDERATIONS. 

PAGE 

The  Molecular  Hypothesis  S. 1 

Dalton's  Contribution 2 

Similarity  of  Molecules      -*  .        . 4 

Hypothesis  of  Avogadro 5 

Distinction  between  Molecules  and  Atoms  .        .        .        .        .        .6 

Classification  of  Bodies 10 

Molecular  Constitution  of  Solids  -T 12 

Molecular  Constitution  of  Liquids^ 12 

Molecular  Constitution  of  Gases  f  14 


Vri.     THE  KINETIC  THEORY  OF  GASES. 

Preliminary  Remarks    .        ...        .        .        .         .         .        .        .14 

Molecular  Collisions  and  Free  Path      .        .         .        .        .        .        .15 

The  Cause  of  Gaseous  Pressure     .        .        .        .  .        .        .17 

Molecules  are  Perfectly  Elastic     .        .        .        .        .        .        .        .17 

Velocities  of  Molecules  Unequal 20 

Statistical  Method  of  Investigation 21 

Fundamental  Assumptions  of  the  Original  Kinetic  Theory         .         .     22 

Maxwell's  Theorem .24 

Illustrations  of  Maxwell's  Theorem .26 

I/-  Determination  of  the  Average  Velocity  of  Translation  of  Hydrogen 

Molecules 29 

^Properties  of  Gaseous  Mixtures 31 

Degrees  of  Freedom 32 

I/Generalized  Theorems  .         .        .         .        .         .        .         .        .        .34 

i  'Adaptation  of  the  Foregoing  Equations  to  the  Generalized  Kinetic 

Theory 37 

^Gaseous  Pressure  .  .        .    38 


yi  CONTENTS. 

PAGE 

Recalculation  of  the  Average  Molecular  Velocity  in  Hydrogen  .        .  39 
Pressure  Produced  by  Several  Sets  of  Molecules          .        .        .        .40 

Avogadro's  Law    .        »     • 41 

Boyle's  Law  .        .        .        ... 41 

l-Results  of  the  Kinetic  Theory  compared  with  Results  of  Observation  .  42 

Temperature         .        .    . 43 

Absolute  Zero       . .        .        .46 

Ratio  of  the  Specific  Heats  of  Gases 47 

/Molecular  Attraction  in  Gases 53 

^/Equations  of  Van  der  Waals  and  Clausius 56 

I- Diffusion 58 

Viscosity ' 59 

Experimental  Determination  of  /* 61 

Kinetic  Explanation  of  Viscosity 63 

Free  Path 66 

High  Vacua 68 

The  Radiometer 68 

Crookes's  Tubes    .                                                                                  ,  70 


III.     THE  MOLECULAR  THEORY  OF  LIQUIDS. 

Preliminary  Remarks 74 

*Free  Evaporation .        .  *     .  75 

^Cooling  Effect  of  Evaporation 77 

I  Vapor  Density 77 

^Vapor  Pressure .79 

Ebullition 80 

Critical  Points 82 

Contraction  and  Compressibility 84 

Surface  Tension 86 

Phenomena  of  Films     .        .        .        .        .        .        .        .        „        .87 

Other  Surfa'ce  Phenomena 89 

Magnitude  of  the  Surface  Tension 90 

Latent  Heat  of  Vaporization 95 

Investigation  of  the  Work  done  in  bringing  a  Molecule  to  the  Surface  96 
Numerical  Estimation  of  the  Work  done  in  bringing  a  Molecule  to 

the  Surface                                                                                ,  98 


CONTENTS.  Vll 

IV.     THE  MOLECULAR  THEORY  OF  SOLIDS. 

PAGE 

Condition  of  the  Theory 102 

Arrangement  of  the  Molecules  in  Solids 103 

Maxwell's  Views  concerning  the  Molecular  Constitution  of  Solids    .  106 

Sublimation 108 

Dissociation          .        .         .        .        . 110 

Solutions ...  112 

Diffusion      .        .        .        ... 114 

^Osmotic  Pressure •        •        .        .117' 

Electrolysis 120 

Saturation •  121 

Distillation '                                        ...  122 

Supersaturation 123 

Crystals        .        .        . 124 

Bounding  Planes  of  Crystals       .        .        .        .        .        .        .        .  125 

Molecular  Structure  of  Crystals 127 

V.     MOLECULAR  MAGNITUDES. 

Preliminary  Remarks 133 

The  Electrical  Method  for  Finding  the  Aggregate  Volume  of  Mole- 
cules         133 

Aggregate  Volume  of  Molecules,  from  the  Gas  Equation  .        .         .  134 

Remarks  on  the  Foregoing  Results .135 

Molecular  Diameters  by  Clausius's  Equation 138 

Lord  Kelvin's  Electrical  Method 139 

Method  by  Camphor  Movements        .        .        .         .        .         .        .  142 

The  Surface  Tension  Method      .        . 143 

Quincke's  Determination  of  the  Range  of  Molecular  Attraction       .  145 

Other  Methods  of  Investigation 146 

Number  of  Molecules  in  a  Unit  Volume  of  Gas          ....  148 

Illustrations  of  Molecular  Magnitudes 149 

VI.     THE  CONSTITUTION  OF  MOLECULES. 

Preliminary  Remarks 151 

General  Facts  to  be  Explained    . 152 

Dulong  and  Petit' s  Law 154 


Viii  CONTENTS. 

PAGE 

Front's  Hypothesis 159 

Periodic  Law  of  Meyer  and  Mendeleieff 161 

Elastic-Solid  Theory  of  Light .164 

Electro-Magnetic  Theory  of  Light 168 

Provisional  Assumptions  about  the  Constitution  of  Molecules  .        .171 

Rankine's  Hypothesis 173 

Lord  Kelvin's  Vortex  Theory 174 

Dr.  Burton's  Strain-Figure  Theory 180 

Internal  Vibration  of  Molecules 182 

Gravitation..       . 190 

Conclusion 196 


APPENDIX. 

On  the  Integration  of  Certain  Equations  in  the  Text  .  .  .  197 
Rankine's  Method  for  Calculating  the  Ratio  of  the  Specific  Heats 

of  Gases 203 

Plateau's  "  Liquide  Glyc<§rique  "  .  .  .  .'  .  .  .206 
On  the  Thermal  Phenomena  Produced  by  Extending  a  Liquid 

Film  .    209 


MOLECULES  AND  THE  MOLECULAK  THEOKY 
OP  MATTER, 

I.    GENERAL   CONSIDERATIONS. 

The  Molecular  Hypothesis.  —  The  molecular  theory  of 
matter,  of  which  I  shall  speak  to  you  this  evening,  declares 
that  every  mass  of  matter,  however  uniform  and  homogeneous 
and  quiescent  it  may  appear,  is  in  reality  composed  of  separate 
particles,  each  of  which  is  in  rapid  motion.  This  proposition, 
although  it  may  seem  extravagant  and  improbable,  has  been 
forced  upon  us  by  a  great  variety  of  considerations,  some  of 
which  I  shall  indicate  to-night.  It  has  been  maintained,  in 
gne  form  or  another,  by  various  philosophers  for  the  last 
2300  years  ;  but  the  reasoning  of  the  ancients,  on  this  subject 
at  least,  is  so  extremely  subtle  and  nebulous  that  it  has  no 
value  whatever  for  modern  purposes.  Nowadays  the  physi- 
cist requires  us  to  state  our  assumptions  very  clearly,  and  to 
deduce  from  them  their  necessary  consequences.  He  will 
then  compare  these  consequences  with  the  observed  facts,  and 
if  the  two  are  in  perfect  agreement  he  will  accept  our  assump- 
tions provisionally,  and  will  believe  in  our  theory  until 
some  one  can  show  that  we  overlooked  some  absurd  things 
that  could  be  deduced  from  our  premises,  or  until  somebody 
brings  forward  another  theory  that  is  just  as  good  as  ours,  or 
perhaps  better.  Although  this  modern  spirit  of  doubt  is 
rather  hard  on  the  "  man  with  a  theory,"  it  is  nevertheless 
quite  logical.  It  prevents  us  from  being  swamped  by  a 
multitude  of  unsound  theories,  and  enables  us  to  distinguish 
the  grain  from  the  chaff.  You  will  agree  with  me,  therefore, 


2        THE  MOLECULAR  THEORY  OF  MATTER. 

when  I  say  that  the  real,  healthy  growth  of  the  molecular 
theory  of  matter  began  when  attempts  were  made  to  obtain 
(erical  results  from  it. 

Dalton's  Contribution. — If  this  be  admitted,  I  think  we 
may  say  that  the  father  of  the  present  molecular  theory  was 
the  English  chemist,  John  Dalton.  In  the  early  part  of  the 
present  century,  Dalton  called  attention  to  the  fact  that 
when  substances  combine  chemically,  they  do  so  in  certain 
definite  proportions.  His  reasoning  was  something  like  this  : 
In  100  pounds  of  carbon  monoxide  there  are  42.9  pounds  of 
carbon,  and  57.1  pounds  of  oxygen.  In  the  same  weight  of 
carbonic  acid  there  are  27.3  pounds  of  carbon,  and  72.7  pounds 
of  oxygen.  No  particular  relations  are  discernible  among 
these  numbers  ;  but  Dalton  found  that  if  the  same  facts  are 
stated  in  a  different  way,  a  very  remarkable  relation  appears. 
Thus,  suppose  we  calculate  what  weight  of  oxygen  is  com- 
bined with  each  pound  of  carbon  in  the  two  gases.  In  carbon 

monoxide  we  find  that  there  are  -~^  =  1.33  pounds  of  oxygen 

~c^*  «7 

to  each  pound  of  carbon,  and  in  carbonic  acid  we  find  that 

72  7 

there  are  -^-^  =  2.66  pounds  to  each  pound  of  carbon.     One 
^Y.o 

of  these  numbers,  you  will  see,  is  exactly  twice  the  other  ; 
and  we  conclude  that  carbon  can  unite  with  oxygen  in  two 
proportions,  the  quantity  of  oxygen,  per  unit  weight  of 
carbon,  being  twice  as  great  in  one  case  as  in  the  other. 
Dalton  observed  similar  relations  among  other  compounds,* 
and  after  turning  the  matter  over  in  his  mind  he  came  to  the 
conclusion  that  the  facts  could  best  be  explained  by  assuming 
that  matter  consists  of  exceedingly  minute,  indivisible  par- 
ticles or  atoms,  each  of  which  has  a  definite  weight.  When 
two  bodies  combine  chemically,  he  conceived  their  atoms  to 
come  together  in  pairs,  or  in  threes,  or  fours,  according  to  the 

*  Dalton's  theory  first  occurred  to  him,  in  fact,  while  he  was  studying 
the  simpler  compounds  of  carbon  and  hydrogen. 


DALTON'S  CONTRIBUTION. 


3 


compound  formed ;  and  he  devised  symbols  to  represent  the 
various  elementary  bodies  and  their  compounds.  Thus,  in 
his  notation, 


Oxygen      =  Q 
Hydrogen  —  0 


Carbon     =  £ 
Nitrogen  =  0 


As  water  was  the  only  compound  of  oxygen  and  hydrogen 
known  to  him,  he  represented  it  by  the  symbol  O0>  con" 
sidering  that  in  it  the  particles  of  oxygen  and  hydrogen  were 
united  in  pairs.  Taking  the  hydrogen  atom  as  the  unit,  it 
follows  that  the  weight  of  the  oxygen  atom  must  be  8  ;  for 
experiment  shows  that  in  a  given  mass  of  water  ibhere  is  8 
times  as  much  oxygen,  by  weight,  as  there  is  hydrogen. 
Carbon  monoxide  was  represented  by  the  symbol  O0?  an(i 
since  for  each  unit  of  its  oxygen  (by  weight)  this  gas  con- 
tains f  of  a  unit  of  carbon,  it  follows  that  the  atomic  weight 
of  carbon  is  f  of  that  of  oxygen.  Hence  the  weight  of  the 
.carbon  atom  is  6.  Carbonic  acid  was  represented  by  the 
symbol  Q  0  O*  Ammonia,  being  the  only  known  compound  of 
hydrogen  and  nitrogen,  was  represented  by  the  simple  symbol 
00  ;  and  as  experiment  shows  that  ammonia  gas  contains 
4|  times  as  much  nitrogen  as  hydrogen,  the  atomic  weight  of 
nitrogen  must  be  4f.  I  have  given  you  a  general  idea  of  the 
kind  of  reasoning  Dalton  used  ;  but  in  calculating  the  atomic 
weights  I  have  made  use  of  better  experimental  results  than 
were  available  to  him.  A  few  of  his  own  early  determinations 
of  the  atomic  weights  are  given  in  the  following  table  :  * 


ELEMENT. 

ATOMIC  WEIGHT. 

Hydrogen    -    - 

1.0 

Nitrogen      -    - 

4.2 

Carbon   -    -    - 

4.3 

Phosphorus 

7.2 

Oxygen  -    -    - 

5.5 

*  These  were  published  in  1805. 


UNIVERSITY 


4        THE  MOLECULAR  THEORY  OF  MATTER. 

Dalton's  fundamental  conception  was  correct,  although  the 
numbers  that  he  used  to  express  the  atomic  weights  of  the 
elements  were  erroneous,  and  so  also  were  many  of  his 
formulae.  We  agree  with  him,  however,  in  believing  that  the 
so-called  "  atomic  weights  "  of  substances  are  really  the  true 
relative  iveiyhts  of  their  atoms ;  the  weight  of  the  hydrogen 
atom  being  taken  as  unity. 

Similarity  of  Molecules.  —  Dalton  assumed  that  all  the 
molecules  of  any  one  substance  are  alike  ;  but  I  think  this 
ought  not  to  be  admitted  without  some  experimental  evidence. 
Various  methods  for  testing  this  assumption  have  been  pro- 
posed, and  while  none  of  them  are  absolutely  convincing,  the 
general  inference  to  be  drawn  from  them  is,  that  there  is  no 
sensible  difference  among  the  constituent  particles  of  any 
given  substance.  I  will  tell  you  briefly  of  two  of  the  methods 
of  investigation  that  have  been  proposed.  Graham's  method 
consisted  in  passing  pure  hydrogen  gas  through  a  porous 
partition  between  two  vessels.  The  first  part  of  the  gas  that 
came  through  was  collected  and  caused  to  pass  through  a 
second  similar  partition.  The  first  portion  that  came  through 
this  partition  was  caused  to  pass  through  a  third  one,  and  so 
the  process  was  continued  until  the  hydrogen  had  passed 
through  a  considerable  number  of  the  partitions.  The  hydro- 
gen from  the  last  operation  was  compared  with  the  original 
hydrogen,  and  no  difference  between  the  two  could  be  dis- 
tinguished. It  was  therefore  considered  that  this  gas,  at 
least,  is  not  a  mixture  of  dissimilar  particles  ;  because  if  it 
were,  such  a  process  as  I  have  described  could  hardly  fail  to 
make  some  sort  of  a  selection  from  among  them,  and  the  final 
gas  would  then  be  different  from  the  primitive  one.  Another 
and  a  more  convincing  method  of  investigating  the  point  in 
question,  was  tried  by.Stas.*  He  determined  the  atomic 
weight  of  the  same  substance  when  prepared  in  different 

*  Stas,  Unter suchung en  iiber  die  Gese+ze  der  chemischen  Proporlionen. 
(Aronstein's  translation.) 


HYPOTHESIS    OF    AVOGADRO.  5 

ways,  from  different  sources,  and  under  different  conditions 
of  temperature  ;  and  he  found  that  the  results  were  indis- 
tinguishable from  one  another.  He  also  found  the  atomic 
weights  of  the  elements  to  be  the  same,  from  whatever  com- 
pound they  were  determined.  His  work  was  so  accurate  that 
it  is  not  likely  that  a  change  in  the  atomic  weight  of  more 
than  the  hundredth  part  of  one  per  cent  would  escape  detec- 
tion. Since  there  was  no  observable  difference,  we  must  con- 
clude that  the  atomic  weights  of  his  different  samples  were 
all  sensibly  alike  ;  and  this  indicates  that  the  molecules  of 
these  substances  were  all  alike,  because  otherwise  we  could 
reasonably  expect  a  slight  difference  to  be  discernible  when  a 
substance  was  prepared  from  different  sources,  by  different 
methods.  From  these  and  other  investigations,  we  conclude 
that  until  some  data  tending  to  prove  the  contrary  are  pro- 
duced, we  may  reasonably  proceed  on  the  hypothesis  that  all 
the  molecules  of  any  given  pure  chemical  substance  are  iden- 
tically alike. 

v  Hypothesis  of  Avogadro.  —  Soon  after  Dalton's  theory  had 
been  announced,  it  was  observed  that  there  are  simple  volu- 
metric relations  among  gases  when  they  combine.  Thus  it 
was  noticed  that  2  volumes  of  hydrogen  combine  with  1 
volume  of  oxygen,  to  produce  approximately  2  volumes  of 
steam  ;  that  1  volume  of  hydrogen  combines  with  1  volume 
of  chlorine  to  form  2  volumes  of  hydrochloric  acid  gas  ;  and 
so  on.  This  being  the  fact,  it  was  suggested  by  Avogadro  in 
1811,  and  independently  by  Ampere  in  1813,  that  all  gases, 
when  under  the  same  conditions  of  temperature  and  pressure,, 
contain  the  same  number  of  molecules  per  unit  of  volume.  This 
assumption  also  explains  the  observed  fact  that  the  densities  \ 
of  gases  are  proportional  to  their  molecular  weights  ;  for  if  * 
t#!  and  w2  are  the  weights  of  the  individual  molecules  of  two 
gases,  and  -ZVj  and  Nz  are  the  numbers  of  the  molecules  in  a 
unit  volume  of  these  gases,  respectively,  then  the  weights  of 
a  unit  volume  of  the  two  gases  are  JV"i  w1  and  N2  w2,  respec- 


6        THE  MOLECULAR  THEORY  OF  MATTER. 

tively.  Now  the  observed  fact  is,  that  these  quantities  are 
proportional  to  Wi  and  w2 ;  and  hence  we  have  the  proportion 

from  which  it  follows  that  N1  =  N2't  that  is,  it  follows 
that  Avogadro's  hypothesis  is  true.  I  do  not  know  whether 
chemists  at  first  received  this  hypothesis  as  the  expression  of 
an  actual  fact  in  nature,  or  merely  as  a  sort  of  convenient 
working  hypothesis.  However  this  was,  subsequent  inves- 
tigation has  made  it  increasingly  probable,  until  now  we  must 
accept  it  as  an  established  fact. 

Distinction  between  Molecules  and  Atoms.  —  Molecules 
may  be  defined  as  the  smallest  parts  into  which  a  given  sub- 
stance can  be  conceived  to  be  divided,  without  changing  its 
chemical  character.  An  atom  is  not  so  easily  defined.  Up  to 
this  point,  in  fact,  I  have  made  no  distinction  between  a 
molecule  and  an  atom  ;  but  if  we  are  to  accept  Avogadro's 
hypothesis,  it  becomes  necessary  to  make  a  distinction  at 
once.  For  if  you  will  think  about  it  a  moment,  you  will  see 
that  if  one  cubic  inch  of  hydrogen,  containing  n  molecules, 
combines  with  one  cubic  inch  of  chlorine,  also  containing  n 
molecules,  to  produce  two  cubic  inches  of  H  01,  containing  n 
molecules  altogether,  then  the  number  of  molecules  in  each 

7? 

cubic  inch  of  the  H  Cl  gas  is  only  —  ;  whereas  Avogadro's  law 

requires  us  to  assume  the '  existence  of  n  molecules  in  each 
cubic  inch.  It  follows,  therefore,  that  when  the  H  and  the 
Cl  combine,  their  molecules  do  not  simply  unite  in  pairs. 
There  is  no  way,  in  fact,  to  explain  the  observed  facts,  unless 
we  assume  that  the  molecules  of  H  and  Cl  are  both  compound, 
and  that  when  these  gases  combine,  their  molecules  split  in 
two,  half  a  molecule  of  the  one  then  uniting  with  half  a  mole- 
cule of  the  other,  to  produce  a  whole  molecule  of  H  01.  This 
is  made  plainer  by  the  diagrams.  Figs.  1  and  2  represent 
small  and  equal  volumes  of  H  and  Cl,  on  the  assumption  that 
their  molecules  are  simple  ;  and  Fig.  4  represents  the  H  Cl 


DISTINCTION    BETWEEN   MOLECULES    AND   ATOMS.        7 


O       0 


FIG.  1.— HYDROGEN  MOLECULES.  FIG.  2.  — CHLORINE  MOLECULES. 


0    •      •     0 

• 

© 

0 

m 

• 

© 

.  ©       •       © 
© 

• 

© 

•  •  .  •  . 

© 

• 

FIG.  3.  —  THE  FOREGOING  GASES  MIXED. 


0» 


FIG.  4.  — THE  SAME  GASES  COMBINED  (AVOGADRO'S  LAW  VIOLATED). 


8 


THE  MOLECULAR  THEORY  OF  MATTER. 


gas  resulting  from  their  combination.  The  space  occupied 
by  the  H  Cl  gas  is  shown  twice  as  large  as  that  occupied 
by  the  component  gases  separately,  because  we  know  from 
experiment  that  when  these  component  gases  combine,  the 


(^ 


eP 


<•£> 


*   ,  %* 

*  «• 

% 


/  •• 


FIG.  5.  — HYDROGEN  MOLECULES. 


FIG.  6. —CHLORINE  MOLECULES. 


&•> 


FIG.  7.  —THE  FOREGOING  GASES  MIXED. 


volume  of  the  result  is  equal  to  double  the  volume  of  either 
one  of  the  constituents.  But  you  will  see  that  the  number  of 
molecules  per  unit  area  is  only  half  as  great  in  the  H  Cl  as  it 
is  in  either  the  H  or  the  Cl  ;  and  this  constitutes  a  violation 
of  Avogadro's  principle.  Now  if  we  conceive  the  molecules 
of  H  and  Cl  to  be  compound,  as  illustrated  in  Figs.  5  and  6? 


DISTINCTION   BETWEEN    MOLECULES    AND    ATOMS.        9 

we  shall  have  no  such  anomaly.  Fig.  8  represents  the  result- 
ing H  Cl,  and  you  see  that  it  fulfills  Avogadro's  law,  as  well 
as  the  observed  fact  of  occupying  two  volumes.  The  com- 
pound nature  of  the  molecules  of  the  so-called  elementary 
bodies  is  no  mere  logical  figment ;  there  is  direct  experi- 
mental evidence  of  its  truth.  It  is  known,  for  example,  that 
many  substances  in  the  nascent  state,  just  being  set  free  from 
Imeir  compounds,  are  much  more  energetic  in  their  chemical 
^relations  than  they  are  under  other  circumstances  ;  and  it  is 
hard  to  explain  this  fact  by  any  theory  that  assumes  the 
molecules  of  such  substances  to  be  simple.  On  the  other 


* 


9    <P 


FIG.  8. —THE  SAME  GASES  COMBINED  (AVOGADKO'S  LAW  FULFILLED). 

.hand,  the  peculiar  activity  of  nascent  substances  is  quite 
intelligible  if  we  assume  that  at  the  instant  they  are  set  free 
the  parts  of  their  molecules  are  not  combined  with  one 
another  to  form  what  I  may  call  normal  molecules ;  for  then 
the  nascent  substance  could  combine  with  other  substances 
without  first  overcoming  the  internal  forces  that  normally 
bind  the  parts  of  its  own  molecules  together.  In  addition  to 
the  known  activity  of  nascent  substances,  we  have  proof  of  the 
compound  nature  of  molecules  in  the  phenomenon  known  as 
gaseous  dissociation,  which  is  doubtless  familiar  to  you  if  you 
have  studied  the  properties  of  sulphur  and  sal  ammoniac  in 
your  course  in  chemistry.  I  must  caution  you  against  sup- 


10       THE  MOLECULAR  THEORY  OF  MATTER. 

posing  that  these  sketches  are  intended  as  pictures  of  the 
actual  molecules.  They  are  simply  diagrams,  intended  merely 
to  aid  you  in  understanding  how  Avogadro's  law  obliges  us  to 
conclude  that  the  molecules  of  even  the  so-called  elements  are 
really  compound  bodies,  capable  of  breaking  up  into  atoms. 
Perhaps  you  will  understand,  now,  what  an  atom  is.  An 
atom  is  one  of  the  parts  into  which  a  molecule  can  be  divided, 
do  not  positively  assert  that  a  hydrogen  molecule  con- 
sists of  only  two  atoms ;  but  we  know  that  when  it  divides  it 
splits  into  halves,  and  in  the  absence  of  any  evidence  to  the 
contrary  we  assume  it  to  be  di-atomic,  though  we  must  always 
remember  that  future  research  may  require  us  to  admit  it  to 
be  tetratomic,  hexatomic,  or  even  more  complex.  In  general, 
we  assume  for  every  substance  the  simplest  molecular  struc- 
ture that  is  consistent  with  the  observed  facts.  The  facts 
now  known  regarding  hydrogen,  chlorine,  and  oxygen  do  not 
require  us  to  assume  more  than  two  atoms  to  the  molecule  ; 
but  in  some  other  bodies  we  do  have  to  assume  more  than 
two  —  in  sulphur,  for  example,  we  are  required  to  admit  that 
the  molecule  is  at  least  hexatomic.  We  believe  that  the  mole- 
cules of  any  one  chemical  substance  are  all  alike  ;  but  we 
know  that  the  atoms  that  compose  these  molecules  are  not 
alike  unless  the  substance  in  question  is  an  element.  (Indeed, 
we  do  not  positively  know  them  to  be  alike,  even  in  this  case ; 
but  we  assume  them  to  be  so,  in  the  absence  of  any  evidence 
to  the  contrary.)  We  know  of  only  about  70  different  kinds 
of  atoms  ;  but  there  are  as  many  kinds  of  molecules  as  there 
are  chemical  substances  ;  and  it  begins  to  look  as  though 
there  is  no  limit  to  the  number  of  these. 

Classification  of  Bodies.  —  Leaving  the  historical  aspect  of 
the  molecular  theory,  let  us  proceed  to  the  consideration  of 
its  present  state.  We  observe  at  the  outset  that  bodies  are 
divisible,  physically  speaking,  into  a  certain  small  number  of 
classes.  We  may,  for  present  purposes,  consider  them  as 
divisible  into  solids  and  fluids;  fluids  being  further  sub- 


CLASSIFICATION    OF    BODIES. 


11 


divided  into  liquids  and  gases.  This  classification  is  not  all 
that  could  be  desired,  but  it  will  serve.  A  solid  body  may  be 
defined  as  a  body  capable  of  resisting  a  considerable  shearing 
strain  —  a  shearing  strain  being  one  which  tends  to  cause  one 
part  of  a  body  to  slide,  relatively  to  some  other  part.  (See 
Fig.  9,  in  which  S  is  a  surface  which  has  experienced  a  shear- 
ing strain  greater  than  the  material  could  sustain.)  Solid 
bodies  usually  have  considerable  tensile  strength  also,  and 
there  are  other  properties  peculiar  to  them,  with  which  you 
are  doubtless  familiar.  It  is  important  to  note  that  a  solid 
does  not  yield  continuously  to  a  small  deforming  force.  It 
resists  deformation,  and  its  resistance  increases  as  the  deform- 


FIG.  9. —DIAGRAM  ILLUSTRATING  "SHEAR." 

ation  increases.  A  fluid,  on  the  other  hand,  is  a  substance 
having  almost  no  shearing  strength,  and  offering  very  little 
resistance  to  forces  that  tend  to  change  its  shape.  A  fluid 
yields  continuously  to  a  deforming  force,  and  a  force  that 
will  deform  it  at  all  will  deform  it  indefinitely,  so  long  as  it 
is  allowed  to  act.  Considering  the  subdivision  of  fluids  into 
gases  and  liquids,  we  may  say  that  a  gas  is  a  fluid  that  < 
presses  continuously  and  in  every  direction  on  the  walls  of  '• 
the  vessel  containing  it,  and  which  follows  them  indefinitely 
if  they  retreat.  A  gas,  if  left  to  itself,  tends  to  expand 
infinitely  in  every  direction.  A  liquid  is  a  fluid  which  does 
not  follow  the  walls  of  the  containing  vessel  if  they  retreat, 
and  which  has  no  tendency  to  expand  infinitely  if  left  to 


12      THE  MOLECULAR  THEORY  OF  MATTER. 

itself.  You  must  understand  that  all  these  definitions  are  to 
a  certain  extent  ideal.  There  is  no  perfect  solid,  nor  any 
perfect  liquid,  nor  any  perfect  gas.  The  ideal  states  are  not 
realized  in  nature  ;  but  as  most  bodies  approximate  more  or 
less  closely  to  one  of  the  three  states  I  have  described,  we 
are  forced  to  conclude  that  there  are  three  principal  molecular 
conditions  in  which  bodies  can  exist.  We  are  far  from  under- 
standing these  conditions  perfectly,  but  I  shall  try  to  give 
you  a  general  idea  of  them,  so  far  as  we  do  understand  them. 

Molecular  Constitution  of  Solids.  — We  are  to  consider  tHat 
in  solids  the  molecules  have  become  relatively  fixed,  in  some 
way  as  yet  very  imperfectly  understood.  I  do  not  mean  that 
they  are  motionless,  however.  They  may  be,  and  almost 
certainly  are,  in  very  active  motion  ;  but  the  .point  is,  they 
do  not  roam  about  among  one  another.  Each  molecule 
executes  a  sort  of  vibration  or  oscillation  about  a  mean  or 
average  position,  from  which  mean  position  it  does  not 
of  itself  permanently  depart ;  nor  does  it  depart  permanently 
from  this  mean  position  under  the  influence  of  small,  tempo- 
rary external  forces.  There  is  a  stable  equilibrium  of  some 
kind  established,  though  I  cannot  tell  you  much  more  about 
it.  You  must  not  imagine  that  the  vibration  of  a  molecule 
in  a  solid  consists  in  a  simple  periodic  to-and-fro  motion. 
The  oscillations  are  probably  of  an  exceedingly  complicated 
nature.,  The  parts  of  the  molecule  may  be  vibrating  among 
themselves,  and  the  molecule  is  almost  certainly  vibrating  as 
a  whole,  in  such  a  way  that  its  center  of  gravity  describes  a 
tortuous  curve,  while  the  molecule  executes  a  series  of  com- 
plex rotations. 

Molecular  Constitution  of  Liquids,,  —  In  liquids,  the  mole- 
cules no  longer  have  determinate  mean  positions.  They  are 
in  active  motion,  however,  and  in  both  solids  and  liquids  they 
exert  a  powerful  attraction  on  their  neighbors.  We  are  to 
conceive  the  molecules  of  a  liquid  as  crowded  closely  together, 


MOLECULAR    CONSTITUTION    OF    LIQUIDS. 


13 


and  continually  curving  about  under  the  varying  influence  of 
their  mutual  attractions,  each  molecule  winding  its  tortuous 
way  in  and  out  among  the  others  perpetually.  You  will  see 
that  according  to  this  conception  any  given  molecule  in  a 
liquid  may  pass  through  every  part  of  the  liquid  in  the  course 
of  time,  though  its  path  is  so  crooked,  and  so  continually 
bending  back  upon  itself,  that  the  molecule  has  to  travel  an 
enormous  actual  distance  before  it  has  departed  very  far  from 
its  starting  point.  I  have  made  a  purely  imaginary  sketch  to 


FIG.  10.  — ILLUSTRATING  MOLECULAR  MOTION  IN  A  LIQUID. 

illustrate  this  kind  of  motion  (Fig.  10).  It  represents  a 
number  of  molecules,  which,  for  the  time  being,  we  will 
suppose  to  be  fixed.  A  molecule  is  supposed  to  enter  this 
congregation  at  A,  and  the  curve  represents  its  motion  until 
it  leaves  the  system  again  at  B.  I  have  represented  a 
collision  with  the  molecule  P,  the  moving  molecule  rebound- 
ing again  and  continuing  its  course.  This  sketch  does  not 
purport  to  represent  the  actual  path  described  by  a  liquid 
molecule,  and  I  present  it  to  you  only  for  the  purpose  of 
assisting  your  imagination  a  little.  In  the  actual  liquid  all 
the  molecules  are  moving,  and  a  diagram  of  their  true  motions. 


14       THE  MOLECULAR  THEORY  OF  MATTER. 

would  probably  be  so  complicated  that  it  would  be  quite  unin- 
telligible, even  if  we  could  construct  it  —  which  unfortunately 
we  cannot.  I  may  say,  however,  that  collisions  between 
liquid  molecules  are  probably  very  much  more  frequent  than 
the  diagram  indicates,  and  that  the  actual  motions  of  the 
molecules  of  liquids  are  probably  fully  as  complicated  as  they 
are  in  solids. 

Molecular  Constitution  of  Gases.  — We  are  to  conceive  the 
molecules  of  a  gas  as  possessing,  on  the  whole,  a  higher 
velocity  than  those  of  liquids  and  solids.  At  least  we  have 
reason  to  believe  that  the  molecules  of  any  one  gas  have  a 
higher  velocity  than  these  same  molecules  would  have  if  the 
gas  that  they  compose  were  liquefied  or  solidified  by  suitable 
means.  Moreover,  the  molecules  of  gases  are  much  further 
apart,  on  the  whole,  than  the  molecules  of  liquids  and  solids 
are.  In  fact  it  has  been  said,  by  way  of  popular  illustration, 
that  the  molecules  of  gases  are  as  far  apart,  in  proportion  to 
their  size,  as  the  fixed  stars  are.  This  is  certainly  not  true. 
They  are  far  enough  apart,  however,  to  be  out  of  the  range  of 
one  another's  attraction  during  the  greater  part  of  the  time  ; 
or,  at  least,  out  of  the  range  of  any  sensible  attractive  influ- 
ence. Now  a  body  left  to  itself,  and  not  acted  upon  by 
external  forces,  describes  a  straight  line  with  unvarying 
velocity ;  and  hence,  you  will  see,  we  must  conclude  that  the 
molecules  of  gases  describe  paths  that  are  sensibly  straight. 
This  is  a  very  important  conclusion,  and  it  underlies  almost 
all  of  the  reasoning  about  gases  that  I  shall  present  to  you 
this  evening. 

II.    THE   KINETIC   THEORY  OF   GASES. 

Preliminary  Remarks.  —  The  properties  of  gases  are  so 
remarkable,  and  the  laws  to  which  they  are  subject  are  so 
simple,  that  physicists  were  naturally  led  to  believe  that  the 
easiest  way  to  learn  something  about  molecules  would  be,  to 


MOLECULAR  COLLISIONS,  AND  FKEE  PATH.     15 

study  the  gaseous  state  of  matter  first.  A  great  deal  of 
thought  has  therefore  been  expended  upon  gases,  and  we  now 
have  some  fairly  accurate  information  about  their  molecular 
constitution.  The  phenomena  of  liquids  and  solids  are  so 
much  more  complicated,  and  so  imperfectly  understood,  that 
it  will  probably  be  many  years  before  our  knowledge  of  the 
molecular  structure  of  these  bodies  is  at  all  comparable  with 
our  present  knowledge  concerning  gases.  For  this  reason  I 
shall  have  to  devote  a  considerable  part  of  the  evening  to  the 
discussion  of  gases,  and  comparatively  little  of  it  to  liquids 
and  solids. 

Molecular  Collisions,  and  Free  Path.  — When  a  multitude 
of  bodies  are  moving  about  in  the  same  region,  in  every  con- 
ceivable direction  and  with  every  imaginable  velocity,  it  is 
perfectly  certain  that  there  will  be  frequent  collisions  among 
them  ;  and  if  we  are  to  consider  gases  as  aggregates  of 
swiftly  moving  molecules,  we  shall  have  to  introduce  the  idea 
of  molecular  collisions.  The  best  popular  description  of  a 
gas,  according  to  the  received  kinetic  theory,  that  I  have 
heard,  likens  it  to  a  wash-boiler  full  of  furious  bumble-bees, 
the  bees  corresponding  to  the  molecules.  The  analogy  is  not 
perfect,  however,  and  we  must  not  press  it  too  far.  *  The 
molecules  of  a  gas,  as  I  have  said,  are  believed  to  %  about 
in  straight  lines,  in  every  conceivable  direction.  When  two 
molecules  collide,  they  bound  apart  like  rubber  balls,  except 
that  they  are  much  more  perfectly  elastic  than  rubber  balls. 
They  crash  into  one  another  incessantly  and  fly  apart  again, 
only  to  crash  into  others  and  rebound  as  before.  The  space 
described  by  a  molecule  between  two  successive  collisions  is 
called  its  free  path  ;  and  I  may  state  a  fact  to  which  I  have 
already  called  your  attention,  in  the  following  more  exact  ^ 
language  :  In  the  gaseous  condition,  the  average  free  path  of  | 
the  molecules  is  so  great,  in  proportion  to  the  range  of  their 
power  of  sensible  attraction,  that  during  the  greater  part  of 
the  time,  each  molecule  is  practically  uninfluenced  by  its 


16 


THE  MOLECULAR  THEORY  OF  MATTER. 


neighbors.  You  will  observe  that  we  do  not  say  that  a  mole- 
cule has  a  smaller  attractive  power  in  the  gaseous  condition 
than  it  has  in  the  solid  or  liquid  condition.  Such  a  statement 
as  that  would  be  indefensible.  But  if,  at  any  instant,  we 
could  suddenly  arrest  all  the  molecules  of  a  gas,  and  fix  them 
just  as  they  were  at  that  instant,  so  that  we  could  examine 
them  at  our  leisure,  we  should  find  that  the  great  majority  of 


FIG.  11.  — ILLUSTRATING  MOLECULAR  MOTION  rx  A  GAS. 

them  were  too  far  apart  to  exert  sensible  attractive  powers 
on  one  another.  The  diagram  (Fig.  11)  may  assist  you  to 
form  a  proper  conception  of  the  motion  of  a  gaseous  molecule. 
It  represents  the  kind  of  path  one  molecule  might  describe  in 
passing  through  an  aggregation  of  other  fixed  molecules  ;  but 
you  must  be  careful  not  to  infer  anything  about  the  shapes  of 
the  molecules,  nor  about  their  sizes  as  compared  with  their 
average  distance  from  one  another,  either  from  this  diagram 


or   from   the  corresponding  one  (Fig.  10)  already  given  to 
illustrate  the  molecular  motion  in  liquids. 


The  Cause  of  Gaseous  Pressure.  —  If  our  conception  of  a 
gas  is  correct,  it  is  plain  that  those  molecules  which  are  in 
the  outer  parts  of  a  given  mass  of  gas  must  beat  incessantly 
upon  the  walls  of  the  containing  vessel,  flying  back  again 
from  these  walls  in  the  same  way  that  they  fly  away  from 
one  another  after  collisions  among  themselves.  This  being 
the  case,  it  is  plain  that  the  walls  of  the  containing  vessel  are 
in  the  same  condition  as  a  target  against  which  a  perfect 
storm  of  bullets  is  striking  perpetually.  Such  a  storm  of 
bullets  would  tend  to  force  the  target  in  the  direction  in 
which  the  bullets  were  moving  before  collision  ;  and  if  the 
impacts  were  frequent  enough,  they  would  have  an  effect 
upon  the  target  which  could  not  be  distinguished  from  a  con- 
tinuous pressure.  And  if  we  pass,  in  thought,  from  bullets  to 
molecules,  and  from  target  to  retaining  vessel,  we  shall  have 
a  very  good  notion  of  the  cause  of  gaseous  pressure,  as  it  is 
understood  to-day. 

Molecules  are  Perfectly  Elastic.  —  If  gaseous  pressure  is 
really  due  to  molecular  bombardment,  there  is  one  property 
of  molecules  that  we  can  point  out  immediately.  It  is,  that 
they  are  perfectly  elastic.  I  am  not  sure  how  far  the  gentle- 
men present  have  followed  the  study  of  mechanics  and 
physics,  so  at  the  risk  of  telling  you  something  you  could 
much  better  tell  me  all  about,  I  am  going  to  explain  what  I 
mean  by  the  expression  "  perfectly  elastic."  If  we  drop  a 
lead  ball  on  the  floor,  it  does  not  rebound  at  all  ;  and  we 
express  this  fact  by  saying  that  it  is  not  elastic  —  or  by  say- 
ing that  its  elasticity  is  practically  zero.  If  we  drop  a  rubber 
ball  on  an  unyielding  surface,  it  does  rebound,  but  not  to  the 
height  from  which  we  dropped  it.  We  express  this  fact  by 
saying  that  it  is  elastic,  but  not  perfectly  so.  The  rubber  or 
glass  or  lead  ball,  in  its  original  elevated  position,  possesses 


18       THE  MOLECULAR  THEORY  OF  MATTER. 

a  certain  amount  of  potential  energy,  or  energy  of  position. 
As  it  falls  it  loses  this  potential  energy  exactly  in  proportion 
to  its  approach  to  the  floor  ;  and  as  energy  cannot  be  destroyed, 
the  ball  moves  quicker  and  quicker,  so  that  its  kinetic  energy 
increases  just  as  fast  as  its  potential  energy  decreases.  When 
it  reaches  the  floor  it  can  go  no  further  ;  but  it  has  by  that 
time  acquired  considerable  velocity,  and,  consequently,  con- 
siderable momentum;  and  the  upper  parts  of  the  ball  are 
crowded  downward  by  their  momentum,  in  spite  of  the  fact 
that  the  lower  part  of  the  ball  has  come  to  rest.  The  obvious 
result  is,  that  the  ball  is  flattened  out,  something  like  this 
(Fig.  12),  where  the  dotted  line  shows 
the  contour  of  the  ball  at  the  moment 
of  striking  the  floor,  and  the  full  line 
shows  the  contour  at  the  moment  of 
maximum  compression.  The  ball  is 
now  stationary,  and  its  kinetic  energy 
has  entirely  disappeared.  What  has 
become  of  it  ?  Well,  if  the  ball  has 

FIG.  12.— ELASTIC  BALL  STRIK-  no  elasticity  at  all,  its  kinetic  energy 
IXG  A  HARD  SURFACE.          ,          ,,  ,  f  ,  .    , 

has  all  been  transformed  into  energy 

of  other  kinds  —  principally  into  heat.  On  the  other  hand, 
if  the  ball  is  perfectly  elastic,  its  kinetic  energy  has  all 
been  stored  up  in  the  ball  as  internal  potential  energy;*  and 
in  another  instant  it  will  beghi  to  be  given  out  again  as 
kinetic  energy  of  translation,  and  the  ball  will  go  up  in  the 
air  once  more  until  it  reaches  the  exact  height  from  which  it 
fell  —  unless  the  shock  of  the  collision  with  the  floor  has  set 
up  vibrations  in  the  ball,  in  which  case  it  will  not  rise  to  the 
original  height,  but  will  fall  short  of  it  by  an  amount  exactly 
corresponding  to  the  amount  of  its  internal  energy  of  vibra- 
tion. Now  if  molecules  did  not  possess  this  property  of 
perfect  elasticity,  then  every  time  they  collided  with  one 
another,  or  with  the  walls  of  the  containing  vessel,  a  portion 

*  The  floor  is  here  supposed  to  be  perfectly  rigid  and  unyielding,  so 
that  it  does  not  absorb  any  of  the  energy  of  the  ball. 


MOLECULES  ARE  PERFECTLY  ELASTIC.       19 

of  their  kinetic  energy  would  be  dissipated  as  energy  of  some 
other  kind  ;  and  hence  in  the  course  of  time  their  velocities 
would  grow  materially  less.  For  instance,  if  we  had  a  mass 
of  gas  in  a  sealed  bottle  through  which  nothing  was  allowed 
to  pass  in  either  direction  —  not  even  light  nor  heat  —  in  the 
course  of  time  the  molecules  would  come  to  rest,  so  far  as 
any  motion  of  translation  is  concerned,  and  in  the  place  of  a 
gas  we  should  have  a  vacuum  with  a  layer  of  inert  molecules 
on  the  bottom  of  the  bottle.  (Of  course  there  would  still  be 
in  the  bottle  an  amount  of  energy  equal  to  the  primitive 
kinetic  energy  of  translation,  rotation,  and  internal  vibration 
of  the  molecules  ;  but  what  form  this  energy  would  have,  I 
cannot  say.)  \This  would  imply  that  the  gas  had  ceased  to 
exist  as  a  gas  ;  and  since  we  have  no  experimental  evidence 
of  such  a  phenomenon,  it  is  plain  that  our  hypothesis  con- 
cerning the  nature  of  gaseous  pressure  obliges  us  to  admit  the 
perfect  elasticity  of  molecules.  This  is  tlie  first  general 
property  of  molecules  that  we  have  arrived  at  ;  and  in  all 
that  follows,  this  evening,  I  will  ask  you  to  bear  it  in  mind. 
I  ought  to  say  a  word,  before  passing  on,  about  the  possibility^  9 
of  the  transformation  of  kinetic  energy  of  translation  intof  \ 
internal  vibrational  energy.  This  was  hinted  at,  a  moment 
ago,  in  discussing  the  behavior  of  the  perfectly  elastic  ball. 
It  has  been  maintained  by  numerous  eminent  physicists  and 
mathematicians,  that  even  if  the  molecules  were  perfectly 
elastic,  their  kinetic  energy  of  translation  and  rotation  would 
gradually  be  transformed  into  vibrational  energy,  so  that  the 
pressure  of  such  a  molecular  medium  as  we  have  imagined 
would  continually  grow  less,  unless  more  energy  were  sup- 
plied from  without.  This  was  considered  to  be  a  grave  objec- 
tion to  the  kinetic  theory  of  gases.  From  a  mathematical 
analysis  of  the  question,  however,  it  appears  that  such  a 
gradual  increase  in  vibrational  energy  could  not  occur.  I 
will  not  discuss  this  matter  further,  now,  but  will  merely 
quote  to  you  the  present  opinion  of  Lord  Kelvin,  who  for- 
merly believed  that  the  vibrational  energy  of  a  system  of  mole- 


20       THE  MOLECULAR  THEORY  OF  MATTER. 

cules  must  increase  indefinitely,  at  the  expense  of  the  energy 
of  translation  and  rotation.  "  I  now  see,"  he  says,  "  that  the 
average  tendency  of  collisions  between  elastic,  vibrating  solids 
must  be  to  diminish  the  vibrational  energy,  provided  the 
total  energy  per  individual  solid  is  less  than  a  limit  depend- 
ing on  the  shape  or  shapes  of  the  solids  :  and  hence,  as 
nothing  is  lost  of  the  whole  energy,  conversion  of  all  but 
an  infinitesimal  proportion  into  translational  and  rotational 
energy  must  be  the  ultimate  result."  *  Thus  you  will  per- 
ceive, vibrational  energy  is  practically  eliminated  from  the 
problem,  and  we  are  justified  in  omitting  it  from  considera- 
tion, for  the  present.  This  amounts  to  saying,  not  only 
that  molecules  are  perfectly  elastic,  but  also  that  the  mean 
coefficient  of  restitution  of  a  molecule  is  unity. 

Velocities  of  Molecules  Unequal.  —  Another  fact  that 
forces  itself  upon  our  attention  is,  that  whatever  the  veloc- 
ities of  the  molecules  in  a  gas  may  be,  they  cannot  be  all  alike. 
For  if  they  were  alike  at  any  given  instant,  it  is  apparent 
that  the  frequent  collisions  among  them  would  speedily 
destroy  the  equality,  and  we  should  shortly  find  some  of  them 
moving  very  rapidly,  and  others  almost  motionless.  We  shall 
see,  later,  that  the  number  of  collisions  among  the  molecules 
of  a  finite  portion  of  gas  in  a  finite  time,  at  ordinary  pressures 
and  temperatures,  is  prodigious  ;  and  we  should  meet  with 
absolutely  insuperable  mathematical  difficulties  if  we  should 
attempt  to  trace  the  path  of  any  one  molecule  in  its  zig-zag 
course  among  the  others,  with  the  idea  of  determining  its 
individual  velocity  at  any  given  moment.  In  fact,  when  we 
come  to  consider  the  actual  number  of  molecules  in  gases 
under  ordinary  conditions,  and  the  number  of  collisions  that 
occur  in  any  given  time,  I  think  we  shall  almost  be  ready  to 
say  that  it  would  take  the  whole  population  of  the  sidereal 
universe  millions  of  centuries  to  find  out  definitely  what 
would  happen  in  a  single  cubic  inch  of  gas  in  one  second. 

*  Popular  Lectures  and  Addresses,  Vol.  I,  p.  464. 


STATISTICAL   METHOD    OF   INVESTIGATION.  21 

Statistical  Method  of  Investigation.  —  It  would  never  do, 
however,  to  advance  a  theory  and  expect  men  to  accept  it,  if 
it  were  so  complicated  as  to  defy  mathematical  investigation. 
A  method  of  dealing  with  this  problem  has  therefore  been 
devised,  which  we  may  call  the  statistical  method.  We  give 
up,  at  the  outset,  all  idea  of  following  the  molecules  indi- 
vidually, and  regard  gases  as  vast  aggregates  of  moving 
particles,  of  which  aggregates  certain  things  of  a  statistical 
nature  must  be  true.  The  whole  kinetic  theory  of  gases 
therefore  becomes  a  sort  of  department  of  the  theory  of  prob- 
abilities. Before  passing  to  the  consideration  of  the  general 
facts  that  have  been  ascertained  concerning  great  congrega- 
tions of  molecules,  I  want  to  say  a  word  about  this  statistical 
method,  which  is  really  very  beautiful,  from  a  mathematician's 
point  of  view.  The  very  fact  that  there  is  a  statistical  con- 
stancy in  things  is  to  me  a  continual  source  of  wonder.  For 
example,  consider  the  number  of  persons  that  leave  Worces- 
ter on  any  particular  daily  train  —  let  us  say,  on  the  10:13 
train  for  New  York.  You  will  find  that  this  train  does  not 
carry  the  same  set  of  passengers  on  any  two  days,  and  yet 
the  travel  on  it  is  fairly  uniform  in  amount  except  when 
there  is  a  foot-ball  game  in  Springfield,  or  a  Christmas  dinner 
in  prospect  somewhere  else,  or  some  other  pronounced  and 
recognizable  cause  of  disturbance.  There  are  variations,  of 
course,  but  they  are  not  marked.  Now  why  is  it  that  this 
train  is  not  empty  on  some  days,  and  why  do  not  all  these 
people  happen  to  want  to  go  at  once  on  some  other  days  ?  If 
you  can  give  a  good  answer  to  this  seemingly  stupid  question, 
you  will  have  made  an  excellent  beginning  in  the  kind  of 
reasoning  on  which  the  kinetic  theory  of  gases  is  based.  The 
general  principles  on  which  this  theory  is  founded  are  the 
same  as  those  underlying  every  kind  of  statistical  investiga- 
tion; and  you  will  find  that  the  mathematical  formulae 
involved  in  all  such  investigations  are  of  the  same  general 
form. 


22       THE  MOLECULAR  THEORY  OF  MATTER. 

Fundamental  Assumptions  of  the  Original  Kinetic  Theory. 

—  In  the  earlier  mathematical  investigations  of  the  properties 
of  gases,  the  molecules  were  assumed  to  be  exceedingly  small 

—  almost  mere^points ;  and  they  were  assumed  to  be  almost 
infinite  in  number.     They  were,  moreover,  considered  to  be 
hard,  smooth,  spherical,  and  perfectly  elastic,  and  to  exert  no 
influence  on  one  another,  when  not  in  actual  contact.     These 


FIG.  13. —CUBES  IN  COLLISION  IN  VARIOUS  WAYS. 

properties  (except  the  elasticity)  were  assumed,  not  because 
it  was  considered  in  the  least  probable  that  molecules  have 
such  properties,  but  in  order  to  lessen  the  mathematical 
difficulties  involved  in  the  subsequent  analysis  ;  for  these 
difficulties  are  great  enough  to  satisfy  anybody,  even  when 
the  problem  is  made  as  simple  as  possible.  It  was  proposed 
to  make  very  simple  assumptions,  and  then  to  trace  out  the 
results  that  would  follow  from  them,  and  see  how  far  these 


THE   ORIGINAL   KINETIC   THEORY.  23 

correspond  with  the  actual  facts.  This  would  give  some  idea 
of  the  admissibility,  or  inadmissibility,  of  the  premises.  I 
will  try  to  explain  the  reasons  for  making  the  various  assump- 
tions I  have  mentioned.  *  The  molecules  were  assumed  to  be 
spherical,  because  spheres  can  collide  with  one  another  in 
only  one  way ;  whereas  other  bodies  can  collide  in  an  infinite 
number  of  ways,  as  you  will  see  by  considering  the  case  of  a 
pair  of  cubes  (Fig.  13).  iThey  were  assumed  to  be  hard,  in 
order  that  collisions  might  be  considered  as  having  no  sensible 
duration,  and  in  order  that  we  might  not  have  to  consider 
vibrations  in  the  body  of  the  molecule.  %They  were  assumed 
to  be  small,  in  proportion  to  the  space  in  which  they  move, 
in  order  that  the  probability  of  a  collision  in  which  three  or 
more  molecules  should  come  together  at 
once' might  become  vanishingly  small  in 
comparison  with  the  probability  of  a  col- 
lision in  which  the  molecules  come  to- 
gether in  pairs  ;  thus  enabling  us  to  avoid 
the  discussion  of  the  more  complex  col- 
lisions. *  They  were  assumed  to  be  practi- 
cally infinite  in  number,  because  statistical 
conclusions  are  not  exact  when  only  * 
small  number  of  things  are  considered. 

\  They  were  assumed  to  be  devoid  of  the  power  of  attraction  or 
repulsion,  because  the  introduction  of  forces  of  this  kind  would 
greatly  complicate  the  treatment  of  the  problem  ;  and  more- 
over we  have  good  experimental  reasons  for  believing  that  in 
actual  gases  the  effects  of  intermolecular  attraction  are  slight. 

*  Finally,  they  were  assumed  to  be  smooth,  in  order  that  we 
might  not  have  to  take  account  of  the  rotations  that  would  be 
established  if  rough  spheres  should  collide.*  Those  of  you 
who  play  billiards  may  understand  this  better  if  I  say  that 
the  molecules  are  supposed  to  be  so  smooth  that  they  take  no 
"english"  from  one  another  —  the  only  force  acting  between 
them  at  the  instant  of  collision  being  a  radial  force,  as  shown 
in  the  diagram  (Fig.  14). 


24       THE  MOLECULAR  THEORY  OF  MATTER. 

Maxwell's  Theorem.  —  These  premises  being  admitted,  it 
may  be  proved  that  if  the  spheres,  or  ideal  molecules,  are 
once  set  in  motion  with  equal  velocities,  or  with  any  distribu- 
tion of  velocities  whatever,  their  mutual  collisions  will  very 
quickly  bring  them  into  such  a  state  that  the  number  of 
spheres  having  velocities  between  v  and  v  +  dv  will  be 


(1) 


In  this  expression  N  is  the  total  number  of  the  spheres, 
TT  =  3.14159....,  e  =  2.71828....'  (the  base  of  Napierian  loga- 
rithms), and  a  is  a  constant  quantity,  whose  value  cannot 
be  found  unless  we  know  some  further  circumstance  about 
the  motion  of  the  spheres.  If  we  know  their  average  velocity, 
for  example,  we  may  determine  a  in  the  following  way  :  There 
are  dN  spheres  that  have  the  velocity  v  ;  and  hence  the  sum 
of  the  velocities  of  these  dN  spheres  is  v  .  dN,  and  the  sum 
of  the  velocities  of  all  the  spheres  is 


r=  oo 

£.  (velocities)  =  I  v  .  dN 


Substituting  in  this  expression  the  value  of  dN  as  given  by 
equation  (1),  we  have 

V  (velocities)  =  4^=  '  f  e~3  '  v*  dv 
"  VTT  ^ 

Upon  effecting  the  integration  this  gives  us 

X4N       a4         2Na 
(velocities)  =  —=  .  -  =  —  - 
a  VTT     *          VTT 

And  since  the  average  velocity  (which  we  will  represent  by 
FO)  is  one  JVtli  of  the  sum  of  all  the  velocities,  we  have,  upon 
dividing  this  last  expression  by  N, 


or 


MAXWELL'S  THEOREM.  25 

Let  us  next  suppose  that  we  know  the  total  kinetic  energy 
of  the  spheres,  instead  of  their  average  velocity ;  and  let  it 
be  required  to  find  the  value  of  a  in  terms  of  this  energy. 
The  kinetic  energy  of  a  sphere  having  the  mass  m  and  the 
velocity  v,  would  be  -J-  mv2-,  and  there  being  dN  spheres  that 
have  the  velocity  v,  the  sum  of  the  kinetic  energies  of  these 
dN  spheres  will  be  -J-  mv'2.  dN,  and  the  total  kinetic  energy  of 
all  the  spheres  will  be 


k  =  I  $  mv2 .  dN  (3) 

r=0 

Substituting  the  value  of  dN  from  equation  (1),  we  have 
k  = 


(4) 


Effecting  the  integration,*  we  have 

3 

4-'—-       4 


.       2Nm    3a5V7r      3  _r     ,      3Ma2 
k  —         -  .  —  —  =  -  Nina?  = 


where  M  is  the  total  mass  of  the  spheres.  There  is  one 
equation  which  I  should  like  to  prove  to  -you  while  we  are 
discussing  molecular  velocities,  because  we  shall  have  occasion 
to  make  use  of  it  later  on.  If  we  represent  the  average  of 
the  squares  of  the  individual  velocities  of  the  spheres  by  u2, 
we  have 


(5) 


The  total  kinetic  energy  of  the  spheres  is 

mvi2      mv,2  m  2  . 

k  =  ~2L  +  -1^  +  ...=-(v1  *  +  v2  2  +  ...) 

and   we    may   substitute    in    this    expression    the   value   of 
(v?  +  1?22  +  •••)  as  obtained  from  (5).     We  find  that 
_  Nmu*  _  Mu2 
2        :    2 

*  The  process  of  integration  is  given  in  the  Appendix. 


26 


THE  MOLECULAR  THEORY  OF  MATTER. 


Substituting  this  value  of  k  in  the  expression  for  a  as  given 
in  equacion  (4),  we  have 

2u 


Going  back  to  equation  (2),  let  us  replace  a  in  that  equation 
by  the  value  we  have  just  found  for  it.     We  then  have 


(6) 


V6 


7T 


This  equation  is  important,  because  it  enables  us  to  calculate 
the  average  velocity,  VQ,  when  the  "mean-square"  velocity,  u, 
is  given. 

Illustrations  of  Maxwell's  Theorem.  —  If  equation  (1)  be 
integrated  between  the  limits  v  =  0  and  v  =  mV0,  we  shall 
obtain  the  number  of  spheres  whose  velocities,  at  any  given 
instant,  lie  between  0  and  in  VQ  .  The  integral  in  question  is 
a  famous  one,  and,  being  what  mathematicians  call  a  "gamma 
function,"  it  is  not  expressible  in  terms  of  any  other  functions 
with  which  you  are  likely  to  be  familiar.  Let  me  pass  over 
the  process  of  integration,  therefore,  and  give  you  only  the 
results  contained  in  this  table.*  The  first  and  third  columns 
give  the  different  values  of  m  for  which  I  have  computed  the 


TABLE  OP  THE  NUMBER  OF  SPHERES  HAVING  A  VELOCITY  OF 
OR  LESS. 


SPHERES  HAVING 

SPHERES  HAVING 

m 

VELOCITY  m*r0 

m 

VELOCITY  mF0 

OR  LESS. 

OR  LESS. 

l 

4 

.0161 

1 

.5330 

i 

.0368 

2 

.9829 

* 

.1120 

3 

.999958 

1 

.2306 

4 

.9999999926 

I 

.3020 

*  The  process  of  integration  is  given  in  the  Appendix. 


ILLUSTRATIONS    OF    MAXWELL  S    THEOREM. 


27 


integral,  and  the  second  and  fourth  columns  give  the  number 
of  spheres  that  have,  at  any  given  instant,  a  velocity  of  m  V0 
or  less.     (The  "  number  of  spheres  "  is  expressed 
as  a  decimal  fraction  of  the  whole  number  in  the 
medium.)    You  will  notice  that  only  1.61  per  cent 
of  the  spheres  will  have  velocities  as  small  as 
one-fourth  of  the  average  velocity,  and  that  over 
£&^eju3ent  of  them  will  have  velocities  less  than 
twice  the  average  velocity.      The  range  of  the 
velocities  of  the  spheres  will  therefore  be  compar- 
atively small ;  and  only  a  surprisingly  small 
proportion  of  them  will  have  velocities  that 
could  properly  be  called  great,  in  comparison 
with,  the  average  velocity  of  all. 
Less  than  one  in  a  hundred  mil- 
lion will  be  moving  as  fast  as  four 
times  the  average  speed; 
and  I  could  show  you  that 
less   than   one   spherk 
1053  will  be  moving 
as  fast  as  ten  times 
the  average   speed. 
I  have  plotted  for 
you    (Fig.    15)    the 
curve   whose   equa- 
tion is 

4        _n2 

y =    /—  .  c    ^a/ 

and   the   plot   may 

serve  to  give  you  a 

better   idea   of  the 

actual    distribution 

of  the  velocities  of   the  spheres  than  you  could  get  from 

a    study   of   the    integral    itself.      The   abscissae   represent 

velocities,  and  the  ordinate,  for  any  abscissa  Vj  is  propor- 


m 


28       THE  MOLECULAR  THEORY  OF  MATTER. 

tional  to  the  number  of  spheres  having  a  velocity  between 
v  and  v-\-dv.  I  have  marked  the  ordinate  corresponding 
to  the  average  velocity,  F0,  and  those  corresponding  to  -j-  F0, 
1-J-  F0,  and  2  F0.  I  have  also  marked  the  ordinate  correspond- 
ing to  the  quantity  a  which  occurs  in  equations  (1)  and  (7), 
and  you  will  see  that  it  is  the  greatest  ordinate  of  the  curve. 
The  area  included  between  any  two  ordinates  represents  the 
number  of  molecules  whose  velocities  lie  between  the  values 
of  v  that  correspond  to  those  ordinates.  Thus  the  shaded 
area  represents  the  number  of  molecules  whose  velocities,  at 
any  given  instant,  lie  between  F0  and  1-J-  F0.  If  one  ordinate 
were  drawn  at  the  origin  and  the  other  at  infinity,  the  area 
between  them  (which  would  then  be  the  entire  area  of  the 
curve)  would  represent  the  number  of  molecules  whose  veloci- 
ties, at  any  given  instant,  lie  between  0  and  oo  .  In  other 
words,  the  area  of  the  entire  curve  represents  the  whole 
number  of  molecules  present  in  the  gas  under  consideration. 
If  we  should  measure  the  area  of  the  shaded  part  with  a 
planimeter,  and  divide  the  result  by  the  area  of  the  whole 
curve,  the  quotient  would  represent  the  number  of  molecules 
whose  velocities  lie  between  F0  and  1-J-  F0,  expressed  as  a 
fraction  of  the  whole  number  of  molecules  present.  By  pro- 
ceeding in  this  way  we  could  construct  a  table  similar  to  the 
one  I  have  just  given  you,  without  resorting  to  the  somewhat 
laborious  process  of  integrating  equation  (1).  You  will  note 
that  the  curve  approaches  the  axis  of  v  indefinitely,  but  that 
it  does  not  actually  touch  it  except  at  infinity.  It  follows 
that  it  is  possible  for  a  given  sphere  to  have  any  velocity 
whatever ;  but  the  probability  of  the  higher  velocities  is 
vanishingly  small.  In  fact  I  have  told  you  that  there  is 
only  one  chance  in  1053  of  a  given  sphere  having  a  velocity 
as  great  as  ten  times  the  average  velocity ;  and  the  proba- 
bility of  higher  velocities  is  still  smaller,  until  the  probability 
-of  an  infinite  velocity  becomes  zero. 


VELOCITY    OF    HYDROGEN    MOLECULES. 


29 


Determination  of  the  Average  Velocity  of  Translation  of 
Hydrogen  Molecules.  —  It  may  interest  you  to  see  how  the 
average  velocity  of  translation  of  the  molecules  of  a  gas  may 
be  determined.  For  this  purpose  we  shall  assume  for  the 
moment  that  gases  are  really  composed  of  spherical  molecules 
such  as  I  have  described  to  you.  In  that  case  all  the  energy 
the  gas  possesses  must  be  kinetic  energy  of  translation ;  for 
we  have  assumed  that  there  are  no  intermolecular  forces,  and 
no  internal  vibrations;  and  we  have  also  assumed  that  the 
collisions  do  not  give  rise  to  rotations.  We  shall  not  take 
air  for  our  example,  because  air  is  a  mixture  of  different 
kinds  of  molecules  ;  and  we  have  not  yet  considered  the 
properties  of  mixtures.  Let  us  therefore  take  hydrogen. 
At  a  pressure  of  14.7  pounds 
per  square  inch  (or  say  2117 
pounds  per  square  foot),  and 
at  the  temperature  of  melting 
ice,  a  cubic  foot  of  hydrogen 
weighs  .005592  of  a  pound. 
Imagine  this  quantity  of  hy- 
drogen enclosed  in  a  cylinder 
(Fig.  16)  having  precisely  one 
square  foot  of  cross-sectional 
area,  and  an  infinite  length ;  and  conceive  it  to  expand 
indefinitely,  pushing  the  piston  P  before  it,  the  resistance 
of  the  piston  being  so  regulated  as  to  just  allow  the  gas 
to  expand.  Then  if  x  is  the  distance  of  the  piston  from 
the  cylinder  head  at  any  moment,  and  p  is  the  pressure 
exerted  by  the  hydrogen  on  the  entire  piston  at  that  moment, 
the  work  done  by  the  gas  in  pushing  the  piston  a  distance  dx 
is  p.dx  ;  and  the  work  done  in  pushing  it  from  x  =  1  (the 
starting  point)  to  x  =  GO  is 


FIG.  16.  —  ILLUSTRATING  ADIABATIC 
"EXPANSION. 


Work  =   I  p  .  dx. 


(8) 


Now  if  we  want  to  find  out  how  much  energy  is  in  the  gas, 


30       THE  MOLECULAR  THEOEY  OF  MATTEK. 

we  must  not  let  it  either  receive  or  give  up  heat  ;  and  you 
will  find,  when  you  study  thermodynamics,  that  this  condition 
is  expressed  by  the  equation 

pxlAl=C  (9) 

where  C  is  a  constant.  To  determine  C,  let  us  observe  that 
we  know  that  when  x  =  1,  p  =  2117  ;  so  that  the  value  of  C, 
in  this  case,  is  2117,  and  the  adiabatic  equation  becomes 


or 
2117 


Substituting  this  value  of  p  in  (8),  we  have 

00 

Work  =  2117  C^  =  5163  foot  pounds. 

' 


Now  on  the  assumptions  we  have  made,  this  must  be  equal  to 
the  united  kinetic  energy  of  the  molecules  ;  or  to 

Nm    (>12  +  ?v>  +  ...)      M 
%m(v?  +  v?  +  ...)=  —  A      r  ^       =  ~2'u 

where  it?  is  the  quantity  defined  by  equation  (5),  and  M  is  the 
total  mass  of  the  gas.  Hence  we  have 

%Mu2  =  5163.  (10) 

Now  the  weight  of  the  hydrogen  under  consideration  being 
.005592  of  a  pound,  its  mass  will  be  .005592  +  32.2  =  .0001737. 
Substituting  this  for  M  in  equation  (10),  and  solving  for  u, 
we  have 

u  =  7710,  and  F0*  =  7103  feet  per  second. 

You  will  please  notice  that  it  is  not  claimed  that  this  is  the 
average  velocity  of  hydrogen  molecules.  All  that  can  be  said 
of  it  is,  that  it  is  their  average  velocity  provided  the  assump- 
tions about  them  that  I  described  to  you  a  few  moments  ago 

*  See  equation  (6). 


PROPERTIES    OF    GASEOUS    MIXTURES.  31 

correspond  to  the  actual  facts.  I  shall  shortly  give  you  a 
better  determination  of  the  velocity  of  the  molecules  of  a 
gas  —  one  which  is  independent  of  the  assumption  of  any 
particular  form  for  the  molecules. 

Properties  of  Gaseous  Mixtures. — If  the  kind  of  reasoning 
by  which  equation  (1)  was  obtained  is  applied  to  a  medium 
composed  of  a  set  of  ^  spheres  each  with  the  mass  ml9  a  set 
of  N2  spheres  each  with  the  mass  m2 ,  and  so  on,  the  spheres 
in  each  set  being  exactly  alike  and  very  numerous,  and  every 
sphere  being  hard,  smooth,  small,  and  perfectly  elastic,  as 
before,  we  shall  find  that  the  different  sets  will  mix  with  one^i 
another  uniformly,  and  that  the  velocities  in  the  spheres  of! 
each  set  will  be-  distributed  precisely  as  though  the  other  sets* 
were  not  present.  The  most  familiar  example  of  a  gaseous 
mixture,  in  nature,  is  air  ;  and  one  of  the  most  striking  things 
about  air  is  its  constancy  of  composition.  In  a  chemical  com- 
pound we  should  expect  the  proportions  of  the  components  to 
be  constant ;  but  in  a  mere  mechanical  mixture  of  oxygen  and 
nitrogen,  for  instance,  we  might  naturally  expect  to  find  a 
material  difference  in  composition  in  two  samples,  when  one 
is  taken,  say,  on  the  Himalaya  Mountains,  and  the  other  on 
the  shores  of  the  Arctic  Ocean.  The  fact  that  no  such  differ- 
ence in  composition  exists  becomes  particularly  significant 
when  we  know  that  the  kinetic  theory  shows  that  constancy 
of  composition  would  necessarily  result,  if  gases  really  are 
composed  of  small  spherical  molecules  such  as  I  have  described. 
Another  very  important  deduction,  due,  I  believe,  to  Maxwell, 
is  that  in  a  mixture  composed  of  several  sets  of  elastic  spheres, 
the  average  velocities  in  the  different  sets  will  not  be  equal ;  i" 
the  set  in  which  the  molecules  are  heaviest  will  have  the 
smallest  average  velocity ;  and,  in  general,  the  velocities  will 
be  such  that  the  average  kinetic  energy  of  a  molecule  of  one 
set  will  be  precisely  equal  to  the  average  kinetic  energy  of  a 
molecule  of  any  other  set.  This  is  one  of  the  most  remarkable 
propositions  in  the  whole  kinetic  theory  of  gases. 


32 


THE  MOLECULAR  THEORY  OF  MATTER. 


Degrees  of  Freedom.  —  I  am  going  to  tell  you  of  some 
much  more  general  theorems  about  gases,  but  before  doing  so 
I  want  to  explain  what  is  meant  by  the  expression  "  degrees  of 
freedom."  A  particle  constrained  to  move  in  a  given  straight 
line  is  completely  described  when  we  have  stated  its  distance 
from  some  fixed  point  in  that  line.  A  particle  constrained  to 
move  in  a  given  plane  is  completely  described  when  we  have 
stated  its  distance  from  two  fixed,  intersecting  straight  lines 
in  that  plane.  A  particle  in  space  is  completely  described 
when  we  have  stated  its  distance  from  three  fixed,  intersect- 
ing planes.  In  the  first  case  we  say  that  the  particle  has 
one  degree  of  freedom,  because  it  has  only  one  coordinate  ;  in 
the  second  case  we  say  it  has  two  degrees  of  freedom,  because 
it  has  two  coordinates,  either  one  of  which. may  vary  inde- 
pendently of  the  other.  In  the  third  case  we  say  that  the 
particle  has  three  degrees  of  freedom,  because  it  has  three 
coordinates,  any  one  of  which  may  vary,  independently  of 
the  others.  The  Century  Dictionary 
defines  a  degree  of  freedom  as  "an 
independent  mode  in  which  a  body 
may  be  displaced."  The  number  of 
degrees  of  freedom  of  a  body  in  free 
space  can  never  be  less  than  three. 
It  may  be  more  than  three,  however. 
In  fact,  it  must  be  more  than  three, 
if  the  body  is  anything  more  than  a 
mere  particle.  Consider,  for  example, 
a  finite  straight  line,  of  given  length. 
One  end  of  it,  say  A,  has  three  degrees 
of  freedom  —  it  can  be  anywhere  in 
space.  The  other  end,  B,  also  has 
three  coordinates,  but  between  these 
coordinates  and  those  of  A  there  is 
an  equation  expressing  the  fact  that  the  length  of  AB  is 
constant.  By  means  of  this  equation  we  could  eliminate  one 
of  the  six  coordinates,  leaving  only  five  that  are  really  inde- 


FIG.  17.  —  A  RIGID  BODY  IN 
SPACE. 


DEGREES   OF   FREEDOM.  33 

pendent;  and  we  therefore  say  that  such  a  line  as  this  has 
five  degrees  of  freedom.  A  rigid  body  in  space  has  six 
degrees  of  freedom :  it  becomes  fixed  when  three  of  its  points 
are  given.  Consider,  for  instance,  the  points  A,  B,  and  (7,  in 
this  sketch  (Fig.  17).  Each  of  these  points  has  three  coor- 
dinates, making  nine  coordinates  in  all ;  and  among  these 
there  are  three  equations,  expressing  the  constancy  of  the 
distances  AB,  BC,  and  CA.  By  means  of  these  three  equa- 
tions we  can  eliminate  three  of  the  nine  coordinates,  leaving 
onty  six  that  are  really  indepen- 
dent ;  and  hence  we  say  that  rigid 
bodies  have  six  degrees  of  freedom. 
Bodies  that  are  not  rigid  have 
more  than  six  degrees  of  freedom ; 
the  number  of  degrees  that  they 
possess  being  always  determined 
by  the  number  of  independent  FIG.  is. -A  SYSTEM  OF  JOINTED 
coordinates  required  to  fix  them. 

For  example,  a  series  of  n  jointed  rods  has  (2n-}-3)  degree's 
of  freedom.  We  may  perhaps  get  a  clearer  idea  of  the  precise 
significance  of  the  expression  "  degrees  of  freedom  "  by  select- 
ing a  mixed  system  of  coordinates  for  defining  the  position  of 
a  body.  Let  us,  for  example,  again  consider  a  rigid  body  in 
space.  The  center  of  gravity  of  this  body  is  free  to  move 
parallel  to  any  of  the  three  axes  of  reference. .  (This  is  the 
same  thing  as  saying  that  the  center  of  gravity  of  the  body 
is  perfectly  free,  because  any  imaginable  motion  of  it  can  be 
resolved  into  components  parallel  to  x,  y,  and  «.  See  Fig.  19.) 
Moreover,  the  body  is  free  to  rotate  about  three  axes,  parallel 
respectively  to  x,  y,  and  z.  (All  other  rotations  can  be 
resolved  into  component  rotations  about  these  three  axes. 
See  Fig.  20.)  Considering  the  three  translations  and  the 
three  rotations,  we  see  that  a  rigid  body  in  space  has  six 
degrees  of  freedom.  Similarly,  a  geometrical  figure  con- 
strained to  move  in  a  plane  has  three  degrees  of  freedom  : 
its  center  of  gravity  can  move  parallel  to  either  axis,  and 


34 


THE  MOLECULAR  THEORY  OF  MATTER. 


the  figure  itself  can  rotate  about  an  axis  perpendicular  to 
the  plane  in  which  it  is  constrained  to  move.  If  I  have 
made,  the  meaning  of  the  expression  "degrees  of  freedom'' 


FIG.  19.— A  SPHEEE  WITH  THREE 
COMPONENT  TRANSLATIONS. 


FIG.  20.  — A  SPHERE  WITH  THREE 
COMPONENT  ROTATIONS. 


clear  to  you,  we  are  prepared  to  pass  to  the  consideration  of 
the  more  general  theorems  about  molecules  that  I  spoke  of 
a  moment  ago. 

Generalized  Theorems.  —  In  the  earlier  days  of  the  kinetic 
theory  of  gases,  molecules  were  assumed  to  be  spherical,  in 
order  to  avoid  the  tremendous  mathematical  difficulties  that 
would  arise  •  if  any  other  form  were  assumed.  There  are 
excellent  reasons,  however,  for  believing  that  most  molecules 
are  not  spherical ;  and  mathematicians  therefore  turned  their 
attention  to  the  more  general  case  in  which  no  particular 
shape  was  assumed.  You  will  understand  that  the  analysis 
of  this  general  problem  is  very  difficult  ;  but  a  satisfactory 
amount  of  progress  has  been  made  with  it,  and  I  will  tell  you 
what  has  been  discovered,  thus  far.  Let  us  consider  a  medium 
composed  of  any  number  of  sets  of  bodies,  such  that  the 
bodies  belonging  to  each  set  are  exactly  like  one  another, 
though  a  body  belonging  to  one  set  may  be  totally  unlike  a 
body  belonging  to  another  set.  Let  these  bodies  have  any 


GENERALIZED   THEOREMS.  35 

number  of  degrees  of  freedom  (which  number  of  degrees  may 
be  different  in  the  different  sets),  and  let  them  be  acted  on  by 
parallel  forces  (such  as  gravity),  or  by  forces  tending  towards 
fixed  centers,  or  by  internal  forces  (that  is,  forces  acting 
within  the  individual  bodies,  between  their  parts).  Let  all  the 
bodies  be  very  small  in  comparison  with  the  total  space  they 
occupy,  so  that  the  chance  of  their  colliding  three  or  more  at 
a  time  is  practically  nothing.  Moreover,  let  them  be  very 
numerous,  and  let  them  be  perfectly  elastic,  and  let  them  be 
smooth,  so  that  when  they  collide  the  only  force  tending  to 
make  them  rotate  is  that  due  to  normal  impact.  Let  them 
be  set  in  motion  among  one  another  with  any  distribution  of 
velocities ;  and  let  them  be  hard,  but  not  infinitely  so,  the 
force  called  into  play  during  collision  being  very  great,  but 
not  necessarily  infinite  (as  it  would  be  if  the  hardness  were 
infinite) ;  and  let  the  duration  of  a  collision  be  exceedingly 
short,  yet  not  necessarily  zero.  These  are  the  assumptions 
made  by  the  kinetic  theory  of  gases  as  it  exists  to-day.  You 
will  see  that  they  are  vastly  more  general  than  those  I 
described  to  you  in  connection  with  the  earlier  investigations. 
The  conclusions  that  have  been  drawn  from  them  are  as 
follows  :  (1)  After  a  short  time,  the  law  of  distribution  of 
positions  and  velocities  in  each  set  of  the  generalized  bodies 
(or  molecules,  as  we  shall  call  them  henceforth)  will  be  pre- 
cisely the  same  as  it  would  be  if  all  the  other  sets  were 
absent ;  so  that  each  set  behaves  as  a  vacuum  to  all  the  rest, 
so  far  as  the  distribution  of  velocities,  and  the  density  of 
aggregation  of  the  molecules  in  any  given  region,  are  con- 
cerned. (2)  This  law  of  distribution  of  the  velocities  in  each 
set  of  molecules  is  the  same  as  that  given  for  spherical  mole- 
cules in  equation  (1).  (3)  The  average  kinetic  energy  of 
translation  of  the  molecules  of  any  one  set  is  equal  to  the 
average  kinetic  energy  of  translation  of  any  other  set.  (4) 
The  total  kinetic  energy  of  each  set  of  molecules  is  divided  up 
equally  among  the  different  degrees  of  freedom  of  that  set.  This 
last  theorem  is  undoubtedly  the  most  remarkable  proposition 


36       THE  MOLECULAR  THEORY  OF  MATTER. 

about  molecules  ever  enunciated.  It  is  due  to  Boltzmann, 
and  it  seems  not  to  have  met  with,  unqualified  acceptance 
among  mathematicians.  Lord  Kelvin,  even,  says  that  he 
"  never  felt  it  possible  to  believe  in  that  theorem  regarding 
the  distribution  of  energy."  My  opinion  is  worth  little, 
compared  with  his,  yet  it  seems  to  me  that  there  can  be  no 
doubt  about  the  validity  of  the  reasoning  on  which  this 
theorem  is  based,  provided  internal  vibrations  are  excluded 
from  consideration.  It  has  never  seemed  proper,  to  me, 
to  consider  a  possible  mode  of  vibration  as  a  "  degree  of 
freedom " ;  and  I  think  that  it  is  the  extension  of  Boltz- 
mann's  theorem  to  the  vibrational  energy  of  molecules  that 
gives  rise  to  the  objections  of  Lord  Kelvin  and  others.  The 
theorem  does  not  apply  to  single  molecules,  of  course.  Any 
molecule,  selected  at  random,  may  nqt  be  rotating  at  all,  or  it 
may  be  rotating  about  some  axis  and  yet  have  its  center  of 
gravity  stationary,  or  it  may  be  entirely  motionless,  or  it  may 
have  component  motions  of  translation  parallel  to  all  three 
axes,  and  a  rotation  which  is  the  resultant  of  rotations  about 
three  axes,  parallel  to  x,  y,  and  2,  respectively ;  and  if  the 
molecule  is  not  a  rigid  body,  it  may  have  other  motions  also 
—  as  many,  in  fact,  as  it  has  degrees  of  freedom.  But  Boltz- 
mann's  theorem  asserts  that  if  rectangular  axes  be  drawn  in  a 
medium  composed  of  a  multitude  of  flying  molecules,  each 
with  n  degrees  of  freedom,  the  total  kinetic  energy  in  the 
medium  will  be  so  distributed  that  one  nth  of  it  will  be  due  to  •• 
the  velocity-components  that  are  parallel  to  x,  one  nth  to  the 
velocity-components  that  are  parallel  to  ?/,  one  nth  to  those 
that  are  parallel  to  «,  one  nib.  to  the  rotation-components 
whose  axes  are  parallel  to  x,  one  nth  to  the  rotation-compo- 
nents that  are  parallel  to  y,  and  so  on,  one  nth.  of  the  total 
kinetic  energy  of  the  medium  being  due  to  the  sum  of  the 
component  motions  in  each  degree  of  freedom.  I  think  you 
will  see,  now,  why  the  determination  of  the  average  velocity 
of  hydrogen  molecules  that  we  made  a  little  while  ago,  is 
unsatisfactory.  We  considered  only  the  kinetic  energy  due 


THE   GENEKALIZED   KINETIC   THEORY.  37 

to  the  translation  of  the  molecules  ;  and  this  amounted  to 
assuming  that  the  molecules  are  not  set  in  rotation  by  their 
collisions.  We  shall  see,  later,  that  hydrogen  molecules  are 
set  in  rotation  by  their  collisions ;  and  we  shall  find,  in  con- 
sequence, that  the  average  velocity  we  deduced  for  them  was 
too  great. 

Adaptation  of  the  Foregoing  Equations  to  the  Generalized 
Kinetic  Theory.  —  The  extension  of  our  conception  of  mole- 
cules, from  spheres  to  smooth,  elastic  bodies  of  any  form,  does 
not  involve  any  very  radical  changes  in  the  formulae  deduced 
from  the  consideration  of  the  motion  of  the  spheres.  Thus 
equation  (1)  will  still  express  the  number  of  molecules  whose 
velocities  of  translation  lie  between  v  and  v  +  dv,  and  equa- 
tions (2)  will  also  hold  true,  since  they  are  derived  from  (1) 
by  a  process  of  reasoning  which  in  no  way  involves  the  form 
of  the  molecules.  But  we  can  no  longer  consider  the  kinetic 
energy  of  the  molecules  to  be  all  translational.  In  fact,  Boltz-  j 
mann's  theorem  tells  us  that  the  kinetic  energy  is  divided  up 
equably  among  the  different  degrees  of  freedom;  and  as 
translation  involves  only  three  degrees  of  freedom,  it  follows 
that  the  kinetic  energy  of  translation,  in  a  gas,  is  equal  to 

-  .  k ;  where  n  is  the  total  number  of  degrees  of  freedom  pos- 
sessed by  a  molecule  of  the  gas,  and  k  is  the  total  kinetic 
energy  in  the  gas.  Hence,  for  k,  in  equation  (3),  we  must 

O   7, 

write  — ;  and  we  must  make  the  same  substitution  in  (4), 
which  is  derived  from  (3).  Therefore  (4)  becomes 

(ii) 


and        a  =  2. 


. 
4  »     3M 


or  nMo?  ,  „       k 

a  = 


Equation  (6)  remains  unchanged,  since  k  does  not  appear  in  it. 


UNIVERSITY 


38      THE  MOLECULAR  THEORY  OF  MATTER. 

Gaseous  Pressure.  —  I  am  sure  I  have  nearly  exhausted 
your  patience  with  mathematical  formulae  ;  but  there  are  one 
or  two  things  more  to  which  I  want  to  call  your  attention, 
before  passing  on  to  matters  involving  less  mathematics.  By 
considering  the  molecules  as  projectiles  striking  against  the 
walls  of  the  containing  vessel,  we  may  find  out  what  their 
mean  velocity  must  be,  in  order  to  produce  the  observed 
pressure  of  the  gas.  The  advantage  of  this  method  over  the 
one  I  have  already  given  you  in  discussing  hydrogen  gas  is, 
that  it  does  not  require  us  to  make  any  assumptions  concerning 
the  constitution  of  the  molecules,  except  the  very  general  one 
that  their  elasticity  is  perfect  —  or,  more  correctly  speaking, 
that  their  average  "coefficient  of  restitution"  is  unity.  With- 
out going  through  with  the  actual  calculation  (which  would 
involve  theorems  in  mechanics  that  you  have  probably  not 
yet  studied),  let  me  say  that  if  we  confine  our  attention  to 
one  kind  of  gas  —  that  is,  to  a  gas  whose  molecules  are  all 
alike,  each  having  n  degrees  of  freedom  —  and  if  we  assume 
that  there  are  no  forces  except  those  due  to  collisions,  then 
the  expression  for  the  pressure  against  a  unit  area  of  the 
containing  vessel  conies  out 

' 


where  A  is  the  absolute  density  of  the  gas,  M  is  its  total  mass, 
and  k  is  its  total  kinetic  energy.  Now  if  the  velocities  of 
translation  of  the  individual  molecules  are  vi9  vZ)  v8,...,  and 
m  is  the  mass  of  a  single  molecule,  then  the  kinetic  energy  of 
translation  of  the  system  will  be 


Substituting  the  value  of  the  parenthesis  as  obtained  from 
equation  (5),  the  expression  for  the  kinetic  energy  of  trans- 
lation becomes 

-  Nmu*,  or  -  Muz, 

z  & 


AVERAGE   MOLECULAR   VELOCITY   IN   HYDROGEN.     39 

•  . 

since  M=  Nm.  Translation  involves  only  three  degrees  of 
freedom;  and  hence,  by  Boltzmann's  theorem,  we  have  the 
proportion 


or  1          2 

K  =  -  nMu. 
6 

Substituting  this  value  of  k  in  (12),  we  have 


From  this  equation  we  can  calculate  the  value  of  u  when  we 
know  the  pressure  and  density  of  a  gas  ;  and  having  found  u, 
we  can  calculate  V0  by  means  of  equation  (6). 

Recalculation  of  the  Average  Molecular  Velocity  in 
Hydrogen.  —  A  cubic  foot  of  hydrogen  at  32°  Fahr.  and  under 
atmospheric  pressure  (i.e.)  2117  pounds  to  the  square  foot) 

weighs  .005592  of  a  pound.     Hence  its  mass  is  '  ,  or 

.0001737.  Substituting  this  for  A  and  2,117  for  p,  in  (13), 
we  have 


.0001737 
Then,  from  equation  (6),  we  have 

F0  =  6,047  X  .9213  =  5,571  feet  per  second, 

which  is  the  average  velocity  of  hydrogen  molecules,  under 
the  given  conditions.  If  two  gases  have  the  same  pressure, 
then,  by  (13), 


And  as  equation  (6)  shows  that  the  average  velocity  of  the 
molecules  of  a  gas  is  proportional  to  u,  it  follows  from  (14) 
that  in  any  two  gases  having  the  same  pressure,  the  average 


40 


THE  MOLECULAR  THEORY  OF  MATTER. 


molecular  velocities  are  inversely  proportional  to  the  square 
roots  of  the  densities  of  the  gases.  This  enables  us  to  calcu- 
late the  molecular  velocities  in  other  gases  very  readily,  when 
the  molecular  velocity  of  any  one  gas  is  known.  By  the  help 
of  equation  (14)  and  the  velocity  we  have  just  obtained  for 
hydrogen,  I  have  calculated  the  following  table  of  the  average 
molecular  velocities  in  several  familiar  gases. 


TABLE  OF  THE  AVERAGE  MOLECULAR  VELOCITIES  OF  GASES,  IN  FEET 
PER  SECOND,  AT  32°  FAHR.,  AND  ATMOSPHERIC  PRESSURE. 


GAS. 

DENSITY. 
(H  =  l) 

AVERAGE 
VELOCITY. 

Hydrogen 

1.00 

5,571 

Oxygen 

15.90 

1,394 

Nitrogen 

14.01 

1,488 

Carbonic  Oxide 

13.96 

1,491 

Carbonic  Acid 

21.94 

1,189 

Pressure  Produced  by  Several  Sets  of  Molecules.  —  I  have 
told  you  that  the  mathematical  investigation  of  the  general- 
ized molecules  described  a  few  moments  ago,  shows  that  if 
there  are  several  sets  of  such  molecules  flying  about  in  the 
same  field,  the  distribution  of  each  set,  and  the  distribution 
of  velocities  in  each  set,  will  be  the  same  as  though  the  other 
sets  were  not  present.  It  follows  from  this  that  the  pressure 
on  the  bounding  walls,  produced  by  all  the  sets  together,  will 
be  equal  to  the  sum  of  the  pressures  that  the  several  sets 
would  produce,  if  each  existed  in  the  same  space  alone.  You/ 
will  probably  recognize  this .  as  the  equivalent  of  Dalton's 
law,  which  states  that  in  a  mixture  of  gases  the  resulting 
pressure  is  the  sum  of  the  partial  pressures  due  to  the 
several  constituent  gases.  Another  way  of  stating  this  law 
is,  In  a  mixture  of  gases,  each  behaves  like  a  vacuum  to 
all  the  rest 


AVOGADRO'S    LAW.  41 

Avogadro's  Law.  —  Going  back  to  equation  (12),  let  «£' 
confine  our  attention  for  the  moment  to  a  unit  volume  of  gas. 
In  this  case  M  becomes  identical  with  A  ;  for,  by  definition, 
the  absolute  density  of  a  gas  is  its  mass  per  unit  volume. 
Hence  (12)  becomes 

' 


Now  if  two  gases  have  the  same  pressure,  that  is,  if  Pi= 
we  have,  from  (15), 


From  this  equation  it  also  follows  that 

O  KI          O  K% 

MI         w2  ' 

But  we  have  seen  that  either  member  of  this  equation  repre- 
sents the  kinetic  energy  of  translation  of  the  molecules  of  the 
corresponding  gas.  Hence  it  follows  that  if  two  gases  have 
the  same  pressure,  they  will  have,  per  unit  of  volume,  the 
same  kinetic  energy  of  translation.  Now  if  k'  represents  the 
kinetic  energy  of  translation  per  molecule;  and  N  is  the  num- 
ber of  molecules  per  unit  volume,  then  we  may  express  this 
last  fact  thus  :  If  two  gases  have  the  same  pressure,  then 


So  that  if  k\  =  k'2,  then  NI  =  N2  .     Hence,  finally,  we  may 
say  that  if  any  two  gases  have  the  same  pressure,  and  the") 
same  kinetic  energy  of  translation  per  molecule,  then  these  V 
gases  will  contain  the  same  number  of  molecules  per  unit  ) 
volume.     If  we  read  "temperature"  in  the  place  of  "kinetic 
energy  of  translation  per  molecule,"  this  statement  becomes 
identical  with  Avogadro's  law. 

Boyle's  Law.  —  Returning  once  more  to  equation  (12),  let 
us  observe  that  the  definition  of  "density"  gives  us,  in  all 
cases,  the  equation 


THE  MOLECULAR  THEORY  OF  MATTER. 


Substituting  this  value  of  A  in  (12),  we  have 


2k   M 


2k 


(16) 


p  =  ——  —  ,  or  pv  =  -- 

nM    v  '  n 

We  may,  for  convenience,  transform  this  equation  thus  : 

Zi      O  K         —        -.  _7  . 


where  k'  is  the  kinetic  energy  of  translation,  per  molecule, 
and  N  (which  does  not  vary  so  long  as  we  confine  our  atten- 
tion to  some  particular  mass  of  gas)  stands  for  the  number 
of  molecules  in  the  gas  under  consideration.  We  are  strongly 
reminded,  by  this  equation,  of  Boyle's  law,  which  states  that 
the  product,  pv,  is  constant  so  long  as  the  temperature  of  the 
gas  does  not  vary  ;  and  if  we  make  the  single  assumption  that 
the  sense-impression  that  we  call  "  temperature  "  is  really  our 
mode  of  perceiving  molecular  kinetic  energy  of  translation,  we 
shall  find  that  the  results  of  the  kinetic  theory  of  gases  cor- 
respond very  closely  with  the  facts  as  actually  observed. 

Results  of  the  Kinetic  Theory  Compared  with  the  Results 
of  Observation.  —  These  correspondences  may  be  exhibited 
as  follows  : 


RESULTS  OF  THE  KINETIC 
THEORY. 

1.  When  two  or  more  sets  of 
molecules  are  put  into  the  same 
region  of  space,  they  diffuse  into 
one  another,  until  the  molecules 
of  each  set  become  uniformly  dis- 
tributed throughout  this  space. 

2.  The  density  of  a  medium 
composed  of  several  sets  of  mole- 
cules   is   equal   to    the    sum    of 
the  densities  the  individual  sets 
would  have,  if  each  existed  sepa- 
rately in  a  space   equal   to  the 
given  space. 


RESULTS  OF  OBSERVATION. 

1.  When   two  or  more  gases 
are  put  into  the  same  vessel,  they 
diffuse   into   one    another,  until 
each  becomes  uniformly  distrib- 
uted throughout  the  vessel. 

2.  The  density  of  a  gaseous 
mixture  is  equal  to  the  sum  of 
the   densities   of   its   component 
gases. 


TEMPERATURE. 


43 


KESULTS  OF  THE  KINETIC 
THEORY. 

3.  The  pressure  on  the  bound- 
aries, due  to  a  medium  composed 
of  several  sets  of  molecules,  is 
equal  to  the  sum  of  the  partial 
pressures  due  to  its  constituent 
sets. 

4.  In    a    molecular     mixture 
there   is   one   physical    quantity 
which  is  the  same  for  every  set 
of   molecules ;  and  that   is,  the 
average  kinetic  energy  of  transla- 
tion, per  molecule. 

5.  If  two  molecular  aggregates 
exert  the  same  pressure  on  their 
containing-walls,   and   have   the 
same  kinetic  energy  of  translation, 
per  molecule,  then  they  will  also 
contain    the    same    number    of 
molecules  per  unit  of  volume. 

6.  In   any  given    mass   of   a 
molecular  aggregate,  the  product 
of    the     pressure     and    volume 
is   proportional    to   the   average 
kinetic  energy  of  translation,  per 
molecule. 


RESULTS  OF  OBSERVATION. 

3.  The   pressure  on  the   con- 
taining-vessel,  due  to  a  gaseous 
mixture,  is  equal  to  the  sum  of 
the  partial  pressures  due  to  the 
constituent  gases  (Dalton's  law). 

4.  In  a  gaseous  mixture  there 
is  one  physical   property  which 
must  be  the  same  for  each  of  the 
constituent   gases;  and  that   is, 
the  temperature. 

5.  If  two  gases  exert  the  same 
pressure  on  their  containing-ves- 
sels,  and  have  the  same  tempera- 
ture, then  they  will  also  contain 
the  same   number  of  molecules 
per  unit  of  volume  (Avogadro's 
hypothesis).* 

6.  In  any  given  mass  of  gas, 
the  product  of  the  pressure  and 
volume    is   proportional   to    the 
absolute    temperature.     (This    in- 
cludes the  laws  of  Boyle,  Charles, 
and  Gay  Lussac.) 


Temperature.  —  It  is  evident  that  there  is  a  sufficient 
agreement  between  the  properties  of  actual  gases,  and  those 
of  the  ideal  molecular  medium  we  have  considered,  to  make 
it  very  probable  that  gases  have  some  such  constitution  as 
we  have  imagined  the  molecular  medium  to  have.  At  all 
events  we  have  found,  as  yet,  no  contradictions.  ~You  may 
not  be  ready  to  admit,  however,  that  the  assumption  made 

*  I  have  included  Avogadro's  hypothesis  among  the  "observed  prop- 
erties," because  it  was  inferred  from  observation  before  it  was  deduced 
from  a  study  of  the  motions  of  discrete  elastic  particles. 


44       THE  MOLECULAR  THEORY  OF  MATTER. 

with  regard  to  temperature  is  an  admissible  one.  The  dis- 
cussion of  this  point  belongs  properly  to  thermodynamics,  but 
since  it  has  a  close  bearing  on  molecular  theories,  you  may 
allow  me  to  say  a  few  words  about  it.  We  all  know  what  is 
meant  when  we  say  that  one  body  is  hotter  or  colder  than 
another  —  we  refer  to  certain  sensations  that  would  be 
experienced  if  we  should  touch  the  bodies,  or  come  very  near 
to  them ;  but  it  is  quite  a  different  thing  to  devise  a  rigid 
scale  that  will  enable  us  to  measure  differences  in  temperature 
in  such  a  way  that  we  can  say  that  a  difference  of  10°,  for 
instance,  on  one  part  of  the  scale,  is  equivalent,  in  some  sense, 
to  a  difference  of  10°  on  any  other  part  of  it.  In  order  to 
devise  such  a  scale  we  shall  have  to  fall  back  on  some  general 
principle  or  law;  and  our  "temperature  sense"  does  not 
furnish  us  any  such  principle  —  at  least,  not  with  any  that  is 
sufficiently  exact  for  scientific  purposes.  It  is  necessary, 
therefore,  to  seek  for  some  such  principle  in  the  world  out- 
side of  ourselves,  and  to  define  "temperature"  arbitrarily,  so 
as  to  make  it  to  conform  to  that  principle  ;  always  remember- 
ing, of  course,  that  the  scale  of  temperature  finally  selected 
must  be  such,  that  measures  obtained  by  means  of  it  shall  not 
be  perceptibly  inconsistent  with  the  crude  observations  we 
can  make  directly,  by  means  of  our  temperature  sense.  The 
commonest  form  of  thermometer  consists  of  a  glass  vessel 
containing  mercury.  It  is  graduated  by  immersing  it  in  the 
steam  arising  from  boiling  water,  and  in  a  mixture  of  ice  and 
water,  marking  the  points  at  which  the  mercury  stands  under 
these  circumstances,  and  dividing  the  space  between  these 
marks  into  (say)  100  equal  parts,  which  are  called  degrees. 
The  scale  so  formed  is  perfectly  arbitrary,  since  it  involves 
whatever  peculiarities  of  expansion  mercury  may  have ;  and 
these  peculiarities  cannot  be  investigated,  without  reasoning 
in  a  circle,  unless  we  can  find  some  kind  of  an  absolute  scale 
of  temperatures  which  shall  be  independent  of  the  properties 
of  any  particular  substance.  The  mercury  thermometer  does 
not  contradict  our  senses,  it  is  true,  but  neither  would  ther- 


TEMPERATURE.  45 

mometers  made  with  a  host  of  other  liquids ;  and  yet  the 
thermometers  made  with  these  other  liquids  would  not  agree 
among  themselves,  nor  with  the  mercury  thermometer. 
(Water  is  inadmissible  as  a  thermometric  fluid,  for  low  tem- 
peratures at  any  rate,  because  near  the  freezing  point  it  gives 
readings  that  contradict  the  direct  evidence  of  the  senses.) 
Now,  I  am  not  going  to  take  you  into  the  mazes  of  ther- 
mometry.  I  wanted  to  call  your  attention  to  the  fact  that 
"  temperature  "  is  not  such  a  definite  conception  as  one  would 
be  apt  to  imagine  unless  he  had  thought  it  over  carefully, 
and  that  a  thermometer  scale  is  an  arbitrary  thing.  I  thought 
this  might  make  it  easier  for  you  to  admit  that  "temperature'7 
may  be  the  sense-impression  that  corresponds  to  molecular 
kinetic  energy  of  translation.  Yet  I  should  not  want  to  leave 
you  with  the  impression  that  heat  measurement  is  not  an 
exact  science.  Let  me  add,  therefore,  to  what  I  have  said, 
that  Lord  Kelvin  has  provided  us  with  what  he  calls  an 
absolute  thermometric  scale,  which  is  quite  independent  of  the 
properties  of  any  particular  body.  He  obtains  this  scale 
from  thermo-dynamical  considerations,*  and  although  his 
definition  of  temperature  is  quite  as  arbitrary  as  any  other 
definition  of  it,  it  is  the  only  one  yet  proposed  that  rests  on 
a  thoroughly  scientific  basis.  For  the  purpose  of  fixing  the 
size  of  his  degrees,  he  defines  the  difference  in  temperature 
between  boiling  water  and  melting  ice  to  be  100°;  and  he 
then  finds  that  the  temperature  of  melting  ice,  on  his  "  abso- 
lute scale,'7  is  273.1°.  Furthermore,  he  finds  that  the  readings 
of  the  air  thermometer  are  almost  identical  with  those  of  his 
absolute  scale,  provided  allowance  is  made  for  the  difference 
of  273.1  Centigrade  degrees  that  exists  between  the  zero  of 
the  absolute  scale  and  the  zero  of  the  ordinary  Centigrade 
scale.  I  have  made  out  a  table,  here,  giving  the  corrections 
to  the  readings  of  several  kinds  of  thermometers,  to  reduce 
them  to  their  equivalents  on  Lord  Kelvin's  absolute  scale. 
In  all  but  the  last  column  the  readings  are  supposed  to  have. 
*  See  the  article  Heat,  in  the  Encyclopaedia  Britannica. 


46 


THE  MOLECULAR  THEORY  OF  MATTER. 


been  previously  corrected  for  calibration,  expansion  of  the 
glass,  error  of  the  fixed  points,  etc.,  and  the  table  gives  only 
the  correction  that  is  made  necessary  by  the  imperfection  of 
the  thermometric  fluid  itself.  The  last  column  gives  the 
corrections  to  be  applied  to  a  certain  crown-glass,  mercurial 
thermometer  that  was  investigated  by  Regnault.  The  cor- 
rections in  this  column  are  far  smaller  than  those  in  the 
preceding  one,  because  they  include  the  correction  due  to  the 
expansion  of  the  glass,  and  the  glass-expansion  and  mercury- 
expansion  corrections  are  of  opposite  sign. 

TABLE    OF   CORRECTIONS,   FOR  REDUCING   THERMOMETER  READINGS  TO 
THE  ABSOLUTE  SCALE. 


HEADING  OF 
THERMOMETER. 

(Centigrade  Degrees.) 

THERMOMETRIC  SUBSTANCE. 

AIR. 

(Constant  Volume.) 

AIR. 

(Constant  Pressure.) 

MERCURV. 

(Alone.) 

MERCURY  AND 
CROWN  GLASS. 

0° 

o°.oo 

o°.oo 

o°.oo 

20 

-  .03 

-  .04 

+  0  .20 

40 

-  .04 

-  .05 

+  0  .29 

60 

-  .04 

-  .05 

+  0  .30 

80 

-  .02 

-  .03 

+  0.20 

100 

.00 

.00 

0  .00 

o°.oo 

120 

+  .03 

+  .03 

-0.29 

+  .08 

140 

•      +  .06 

+  .07 

-0  .70 

+  .21 

160 

+  .10 

+  .11 

-1  .19 

-1-  .36 

180 

+  .14 

+  .16 

-1  .80 

+  .51 

200 

+  .18 

+  .20 

-2  .49 

+  .48 

220 

+  .22 

+  .25 

-3  .29 

+  .42 

240 

+  .27 

+  .29 

-4  .17 

+  .37 

260 

+  .31 

+  .34 

-5  .15 

+  .11 

280 

+  .36 

+  .39 

-6  .23 

-  .16 

300 

+  .41 

+  .44 

-7  .39 

-  .67 

Absolute  Zero.  —  One  of  the  most  interesting  things  about 
the  absolute  scale  is,  that  it  has  a  zero  —  usually  called  the 
absolute  zero  —  below  which  it  appears  to  be  impossible  to 


RATIO    OF    THE    SPECIFIC    HEATS    OF    GASES.  47 

cool  things.  You  will  note  the  bearing  of  this  on  the  kinetic 
theory  of  gases  ;  for  if  temperature  is  really  our  mode  of 
perceiving  the  translatory  kinetic  energy  of  molecules,  then 
if  we  should  gradually  abstract  from  a  gas  its  translatory 
kinetic  energy,  its  temperature  would  seem  to  fall  lower  and 
lower,  until  finally,  when  we  had  abstracted  all  of  this  energy, 
the  temperature  would  reach  a  point  lower  than  which  it 
could  not  go.  There  would  no  longer  be  any  translatory 
kinetic  energy  to  perceive  ;  and  hence  we  should  have  reached 
an  absolute  zero,  below  which  it  is  not  thinkable  that  the  gas 
could  be  cooled.  I  may  say,  in  fact,  that  if  we  define  temper- 
ature as  Lord  Kelvin  defines  it  (and  this  is  the  only  rational 
way  yet  proposed),  and  if  we  admit  that  the  sense-impression 
that  we  call  "  temperature "  is  our  mode  of  perceiving  the 
kinetic  energy  of  translation  of  molecules,  then  the  kinetic 
theory  of  gases  becomes  even  more  remarkable  than  we  have 
found  it  to  be  ;  because  we  can  then  deduce  from  it  all  the 
fundamental  equations  of  thermodynamics. 

' 

Ratio  of  the  Specific  Heats  of  Gases.  —  If  a  certain  mass 
of  gas,  having  a  volume  v0  at  the  pressure  p0,  expands 
adiabatically  —  that  is,  without  either  receiving  or  giving 
out  heat  as  heat  —  we  know  from  thermodynamics  that  its 
pressure  and  volume  are  connected  by  the  relation 

p0v0y=pvy  (18) 

where  y  is  the  ratio  of  the  specific  heat  of  the  gas  at  constant 
pressure  to  its  specific  at  constant  volume.  Now  if  this  given 
mass  of  gas  is  allowed  to  expand  indefinitely,  or  until  its 
volume  becomes  infinite,  the  total  amount  of  work  it  can  do  is 


Work 


oo 

=  fp  .  dv:  (19) 


If  we  substitute  in  this  equation  the  value  of  p  as  obtained 
from  (18),  and  perform  the  integration,  we  have 


(20) 


48       THE  MOLECULAR  THEORY  OF  MATTER. 

Now  the  work  that  a  gas  can  do  under  such  circumstances 
(assuming  that  there  are  no  forces  between  the  molecules) 
is  equal  to  its  total  kinetic  energy,  kQ.  Equating  k0  to  the 
second  member  of  (20),  we  have 

P»v0=(y-l)k0,  (21) 

and  comparing  this  with  equation  (16)  we  see  that 

y  —  l  =  -,     or     y  =  l  +  ?«  (22) 

n  n 

From  this  equation  we  can  calculate  the  ratio  of  the  specific 
heats  of  a  gas,  if  we  know  n,  the  number  of  degrees  of  freedom 
of  its  molecules  ;  and  conversely,  we  can  calculate  from  it  the 
number  of  degrees  of  freedom  of  the  molecules,  if  we  know 
the  ratio  of  the  specific  heats  of  the  gas.  Now  the  smallest 
value  that  n  can  have  is  three  ;  for  it  must  take  at  least  three 
coordinates  to  fix  the  position  of  a  body  in  space.  The  value 
n  =  3  corresponds  to  the  case  in  which  the  molecules  are 
smooth  spheres,  incapable  of  being  set  in  rotation  by  their 
mutual  impacts.  Mercury,  for  chemical  reasons,  is  believed 
to  contain  only  one  atom  in  its  molecule  ;  and  hence  it  will 
be  interesting  to  see  whether  the  ratio  y,  calculated  on  the 
hypothesis  of  a  smooth,  spherical  molecule,  corresponds  with 
the  actual  value  of  this  ratio  for  mercury  vapor.  We  find, 
for  n  =  3,  y  =  If  —  1.666  ;  and  the  ratio  of  the  specific  heats 
of  mercury  vapor,  as  determined  experimentally  by  Kundt  and 
Warburg,  is  1.66.  The  agreement  of  this  with  the  calculated 
value  lends  considerable  plausibility  to  the  supposition  that 
the  molecules  of  mercury  are  smooth  and  spherical,  and, 
incidentally,  to  the  whole  kinetic  theory.  In  the  case  of 
other  gases  the  agreement  is  not  so  satisfactory.  For  example, 
the  molecules  of  hydrogen,  oxygen,  nitrogen,  and  carbonic 
oxide,  are  believed,  for  chemical  reasons,  to  consist  of  two 
atoms.  We  are  led,  naturally,  to  examine  the  results  of  the 
hypothesis  that  their  molecules  each  consist  of  two  smooth, 
spherical  atoms,  rigidly  united  by  attractive  forces  or  other- 
wise. The  number  of  degrees  of  freedom  that  we  have  to 


EATIO    OF    THE    SPECIFIC    HEATS    OF    GASES. 


49 


consider  in  such  a  system  is  5  (the  freedom  to  rotate  about 
the  line  of  centers  of  the  spheres  is  not  considered,  because 
as  the  spheres  are  assumed  to  be  perfectly  smooth  the  impacts 
of  the  molecules  cannot  set  up  rotations  about  this  axis). 
When  n  =  5,  we  have  y  =  1  -f-  f  =  1.400.  The  accepted  values 
of  y  for  these  gases,  as  given  by  experiment,*  are  presented  in 
this  table : 

TABLE  or  VALUES  OF  7. 


GAS. 

EXPERIMENTAL  y. 

CALCULATED  y. 

OXV°"G11 

1.402 

1  400 

Nitrogen     
Hydrogen  

1.411 
1.412 

1.400 
1.400 

Carbonic  oxide    .... 

1.418 

1.400 

At  first  thought  this  seems  like  a  very  satisfactory  agree- 
ment ;  but  it  is  not  so.  We  have  assumed,  in  deducing  equa- 
tion (22),  that  the  effects  of  the  intermolecular  forces  are 
insensible  ;  but  it  can  be  shown  that  if  they  were  sensible  we 
should  have  to  modify  equation  (22)  so  as  to  make  it  read 


where  x  is  a  small  positive  quantity,  vanishing  when  the 
forces  between  the  molecules  are  insensible.  You  will  please 
note  particularly  that  x  is  necessarily  positive  if  the  forces 
are  attractive,  so  that  the  calculated  values  of  y  would  be 
smaller,  if  we  take  these  forces  into  account,  than  it  would  be 
if  we  neglected  them  and  considered  x  to  be  zero  ;  whereas 
the  observed  fact  is,  that  the  values  of  y  are  larger  than  the 
computed  value  obtained  by  making  x  =  0.  This  constitutes 
an  objection  to  the  kinetic  theory,  which  is  worthy  of  serious 
consideration.  Mathematicians  have  endeavored  to  account 


*  These  experimental  results  are  from  the  Encyclopaedia  Britannica, 
article  Steam  Engine. 


50       THE  MOLECULAR  THEORY  OF  MATTER. 

for  the  observed  discrepancy  in  various  ways,  but  without 
any  very  distinguished  success.  You  will  find  a  suggestion 
made  in  Mr.  Watson's  little  book  on  the  kinetic  theory  of 
gases  ;  but  as  it  involves  some  rather  intricate  considerations, 
and  cannot  be  regarded  at  present  as  anything  more  than  a 
suggestion,  I  shall  not  trouble  you  with  it  this  evening.  I 
have  an  idea  that  the  true  explanation  will  be  found  to 
involve  the  consideration  of  gaseous  dissociation.  It  is  known 
that  many  gases  exhibit  this  phenomenon  in  a  marked  degree 
when  their  temperature  is  sufficiently  high.  According  to 
what  I  have  told  you,  raising  the  temperature  of  a  gas  is 
really  the  same  thing  as  accelerating  its  molecules.  When 
the  mean  speed  of  the  molecules  reaches  a  certain  value,  the 
shocks  due  to  the  molecular  collisions  become  so  great  that 
the  internal  attractive  forces  existing  within  the  molecules 
are  no  longer  sufficient  to  hold  them  together.  They  break 
up,  therefore,  into  simpler  molecules,  or  perhaps  into  their 
constituent  atoms.  This  phenomenon  is  known  as  dissociation. 
Owing  to  the  great  variety  of  velocities  that  exist  within 
any  given  mass  of  gas,  the  dissociation  does  not  take  place 
suddenly,  when  the  gas  reaches  a  particular  temperature.  If 
the  temperature  of  the  gas  be  gradually  raised,  there  will 
come  a  time  when  a  considerable  number  of  the  molecules 
possess  the  velocity  requisite  for  dissociation,  although  the 
great  mass  of  them  may  still  have  velocities  that  are  con- 
siderably below  this  critical  value.  Dissociation  then  com- 
mences. If  the  temperature  of  the  gas  be  now  kept  constant 
the  dissociation  does  not  proceed  until  the  molecules  are  all 
split  apart,  because  many  of  the  dissociated  parts,  coming 
together  again  at  velocities  less  than  the  critical  velocity, 
re-combine  and  produce  new  molecules  like  those  of  the 
original  gas.  You  will  see,  therefore,  that  at  any  given 
temperature  dissociation  proceeds  only  until  there  is  an 
equilibrium  established  between  the  molecules  that  are  break- 
ing up,  and  those  that  are  re-forming.  If  the  temperature  be 
now  raised,  the  average  velocity  will  come  nearer  to  the 


KATIO    OF   THE   SPECIFIC    HEATS    OF   GASES.  51 

critical  value,  and  when  equilibrium  has  been  established  at 
this  new  temperature,  the  number  of  molecules  that  exist  in 
the  dissociated  condition  at  any  given  instant  will  be  greater 
than  before.  If  the  temperature  be  high  enough,  the  num- 
ber of  dissociated  molecules  that  happen  to  collide,  during 
any  given  time,  with  velocities  sufficiently  small  to  allow 
of  re-combination,  may  be  so  insignificant  that  we  cannot 
recognize  the  presence  of  these  re-combined  molecules  by  any 
of  the  chemical  or  physical  tests  at  our  command.  The  gas 
is  then  said  to  be  wholly  dissociated.  Now  it  seems  probable 
that  in  any  gas  there  must  be  some  molecules  in  a  state  of 
dissociation,  even  though  the  temperature  may  be  far  below 
the  critical  value  conventionally  called  the  temperature  of 
dissociation  ;  for  we  have  seen  that  in  any  given  mass  of  gas 
there  are  always  some  few  molecules  moving  with  extreme 
velocities  —  velocities  great  enough  to  produce  dissociation. 
We  may  therefore  conceive  hydrogen  gas,  for  example,  to  be 
an  aggregation  of  molecules,  by  far  the  greater  number  of 
which  are  diatomic  with  5  degrees  of  freedom,  but  some  of 
which,  nevertheless,  are  monatomic  with  3  degrees  of  freedom. 
The  value  of  y  for  such  a  gas  would  be  intermediate  between 
the  value  calculated  for  n  —  5  and  that  calculated  for  n  —  3  ; 
or  in  other  words,  it  would  lie  between  1.400  and  1.666,  but 
far  nearer  the  former  value  than  the  latter.  It  seems  to  me 
that  the  calculated  and  observed  values  of  y  can  be  reconciled 
in  this  way,  but  before  we  could  prove  this  to  be  the  fact  we 
should  have  to  make  a  rigid  mathematical  investigation  of  the 
theory  of  dissociation.  If  this  be  the  true  explanation,  it  is 
evident  that  the  value  of  y  must  increase  when  the  tempera- 
ture of  the  gas  increases  ;  for  at  higher  temperatures  there 
would  be  a  greater  proportion  of  dissociated  molecules  present. 
I  know  of  no  experiments  sufficiently  accurate  to  test  this 
point.  There  is,  of  course,  a  possibility  that  the  observed 
values  of  y  are  incorrect,  owing  to  the  existence  of  some 
unrecognized  source  of  error  tending  to  give  results  uniformly 
too  great ;  but  this  would  have  to  be  proved  to  be  the  fact 


52       THE  MOLECULAR  THEORY  OF  MATTER. 

before  we  could  accept  it  as  the  true  explanation  of  the 
diiference  between  calculation  and  observation.  The  ratio  of 
the  specific  heats  of  gases  is  usually  calculated  from  the 
velocity  of  sound  as  observed  in  the  gases ;  and  as  the  neces- 
sary measurements  are  difficult  to  make,  we  find  that  with 
different  data  different  results  are  obtained.  Thus  from 
Dulong's  data  for  dry  air  we  have  y  =  1.410,  while  from  the 
data  for  air  as  given  by  Eegnault  and  Poisson,  we  find 
y  =  1.401.  So  far  as  the  gases  in  the  table  are  concerned,  it 
is  worthy  of  note  that  a  recent  determination  of  y  for  hydro- 
gen (given  by  Clausius  and  quoted  in  Watts's  Dictionary) 
gives  y  =  1.3852.  I  also  find  that  if  the  value  of  y  be  calcu- 
lated by  combining  the  specific  heat  at  constant  pressure  as 
observed  by  Regnault,  with  the  difference  between  the  specific 
heats  as  calculated  by  E-ankine's  method,  we  have,  for  oxygen, 
y  =  1.398  ;  using  Regnault's  results  for  the  density  of  oxygen, 
Thomson's  determination  of  the  absolute  zero,  and  Griffiths's 
value  of  the  mechanical  equivalent  of  heat.*  In  view  of  the 
differences  that  exist  among  the  different  experimental  deter- 
minations of  y,  I  think  it  would  be  unwise  to  conclude  that 
the  present  discrepancy  between  the  kinetic  theory  and  the 
apparent  facts  may  not  be  cleared  up  satisfactorily  in  the 
future.  We  must  note,  however,  that  since  3  is  the  smallest 
number  of  degrees  of  freedom  that  a  free  body  in  space  can 
have,  it  follows  from  (22)  that  y  —  If  is  the  largest  ratio  of 
the  specific  heats  that  the  kinetic  theory  is  capable  of  explain- 
ing. If  it  can  be  shown  that  the  ratio  of  the  specific  heats  of 
any  gas  is  greater  than  this,  it  looks  as  though  the  kinetic 
theory  would  have  to  go.  You  may  be  interested  to  know 
that  for  highly  superheated  steam  the  value  of  y  is  1.30. 
This  seems  to  correspond  to  a  molecule  with  6  degrees  of 
freedom  ;  for  with  this  value  of  n,  equation  (22)  gives 
y  — 1.333.  The  observed  value  of  y  for  carbonic  acid  gas  is 
1.263,  which  seems  to  indicate  7  degrees  of  freedom  ;  for 

*  The  same  method  gives  for  H,  y  =  1.408  ;   and  for  N,  y  =  1.407. 
See  Appendix. 


.    MOLECULAR   ATTRACTION   IN   GASES.  53 

when  n  =  7  equation  (22)  gives  us  y  =  1.286.  These  values 
of  n,  for  steam  and  for  C02,  seem  to  indicate  that  steam 
molecules  are  rigid  bodies,  and  that  the  molecules  of  C02  may 
consist  of  two  smooth  bodies  jointed  together  in  some  way  ; 
but  in  speculating  in  this  manner  on  the  actual  forms  of  the 
molecules  we  are  going  a  good  way  beyond  the  limits  of 
positive  knowledge. 

Molecular  Attraction  in  Gases.  —  Thus  far  we  have  as- 
sumed that  the  molecules  of  a  gas  do  not  attract  one  another, 
but  that  the  phenomena  of  gases  result  from  the  motions  of 
the  molecules,  unrestrained  except  by  their  collisions  with 
one  another  and  with  the  walls  of  the  containing-vessel.  It 
appears,  from  the  agreement  of  the  results  of  this  assumption 
with  the  observed  facts,  that  the  effects  of  the  mutual  attrac- 
tions that  may  exist  between  the  individual  molecules  of  a 
gas  are  small,  on  the  whole.  The  forces,  when  they  exist,  may 
be  great ;  but  since  their  effects  are  scarcely  noticeable,  we 
must  conclude  that  under  ordinary  circumstances  the  sphere 
of  sensible  action  of  these  forces  is  quite  small  in  comparison 
with  the  length  of  the  average  free  path  of  the  molecules. 
We  may  investigate  the  attraction  or  repulsion  that  may  exist 
between  gaseous  molecules,  by  allowing  a  given  gas  to  expand  1 
into  a  vacuum  so  that  it  shall  do  no  external  work.  If  there 
is  an  attraction  between  the  molecules,  the  gas  will  be  cooled; 
for  some  of  its  kinetic  energy  will  be  transformed  into  poten- 
tial energy.  The  experiment  being  performed,  it  is  found 
that  there  is  almost  no  change  in  temperature  produced  by 
simple  expansion,  when  the  gas  does  no  external  work.  Very 
accurate  experiments  by  Thomson  and  Joule,  however,  showed 
that  there  is  a  slight  temperature  change,  though  it  is  so 
small  that  it  could  readily  escape  observation.  In  order  to 
avoid  eddies  and  other  such  sources  of  error,  Joule  and 
Thomson  caused  the  gases  they  experimented  upon  to  flow 
from  one  vessel  into  the  other  through  a  porous  plug.*  The 

*  Encyclopaedia  Britannica,  article  Heat. 


54       THE  MOLECULAR  THEORY  OF  MATTER. 

initial  pressure,  in  their  experiments,  ranged  from  1  up  to 
5  or  6  atmospheres,  the  final  pressure  being  1  atmosphere  in 
every  case  ;  and  the  cooling  effect  was  found  to  be  propor- 
tional to  the  difference  between  the  initial  and  final  pressures. 
In  the  case  of  air,  the  cooling  effect  was  0°.  208  C.  per 
atmosphere.  With  carbonic  acid  gas  it  was  1°.  105  C.  With 
hydrogen,  on  the  other  hand,  there  appeared  to  be  a  heating 
effect  of  °.039  C.  per  atmosphere.  (Owing  to  its  peculiar 
behavior,  Regnault  called  hydrogen  un  gaz  plus  que  parfait  — 
"a  more  than  perfect  gas.")  These  experiments  show  that 
there  are  attractive  forces  between  the  molecules  of  air,  and 
also  between  those  of  carbonic  acid  gas.  In  expanding,  the 
molecules  of  these  gases  have  become  more  widely  separated, 
and  energy  that  was  before  kinetic  —  and  therefore  sensible 
as  heat  —  has  become  potential  in  overcoming  the  attractive 
forces.  The  behavior  of  hydrogen  is  anomalous.  No  other 
gas  exhibits  a  heating  effect  when  expanding  into  a  vacuum. 
The  discussion  of  this  isolated  phenomenon  would  take  so 
much  time  that  I  shall  not  enter  upon  it  this  evening.  We 
must  note  that  the  cooling  effect  observed  by  Joule  and 
Thomson  affords  a  rough  indication  that  the  attraction 
between  gaseous  molecules  does  not  follow  the  inverse-square 
law  with  which  we  are  so  familiar  in  the  laboratory  and  the 
observatory.  It  indicates,  in  fact,  that  the  attraction  varies 
something  like  the  inverse  fourth  power  of  the  distance. 
For  let  us  assume  that  the  law  of  attraction  is 


where  D  is  the  mean  distance  between  two  neighboring  mole- 
cules. Then  if  the  gas  expands  until  this  mean  distance 
becomes  D19  we  may  express  the  work  done  against  the 
attractive  -forces  somewhat  as  follows  : 


'MOLECULAR   ATTRACTION   IN   GASES.  55 

Now,  as  D  is  a  linear  dimension  in  the  gas,  the  volume  of 
the  gas  will  vary  as  the  cube  of  D.     Hence,  we  may  write 
v  =  t)'Ds,  and  vl  =  b  •  A3- 

Substituting,  in  (24),  the  values  of  D8  and  D*,  as  given  by 
this  equation,  we  have 

Work  =  «*/l_l\ 
3  \v      vj 

But  as  the  temperature  of  the  gas  is  practically  constant,  we 
have 


Substituting,  in  (25),  the  values  of  v  and  vl  ,  as  given  by  this 
equation,  we  have,  finally, 


x^> 
That  is,  the  assumed  law  of  attraction*  indicates  that  when^a 

gas  expands  into  a  vacuum,  the  amount  of  work  done  in  over- 
coming the  internal  attractive  forces  —  'or,  what  is  the  same 
thing,  the  amount  of  kinetic  energy  that  disappears  and  ceases 
to  be  sensible  as  heat  —  is  proportional  to  the  difference 
between  the  initial  and  final  pressures  ;  and  this  is  the  1 
observed  fact.  I  am  aware  that  equation  (24)  is  only  a  crude 
expression  for  the  amount  of  internal  work,  and  that  con- 
sequently the  reasoning  I  have  given  you  does  not  prove  that 
the  inverse  fourth-power  is  the  true  law  of  variation  of  inter- 
molecular  attraction  ;  but  I  think  it  is  fair  to  say  that  the 
reasoning  shows,  at  least,  that  when  two  molecules  approach 
each  other,  the  attraction  between  them  increases  faster  than 
the  inverse-square  law  would  indicate.  Although  the  reason- 
ing I  have  given  you  does  not  prove  that  equation  (23)  rep- 
resents the  true  law  of  molecular  attraction,  Mr.  William 
Sutherland  has  recently  shown,  in  a  more  rigorous  manner, 
that  this  is  very  likely  the  fact.f  I  may,  perhaps,  venture 

*See  equation  (23). 

t  See  his  recent  articles  in  the  Philosophical  Magazine. 


56      THE  MOLECULAR  THEORY  OF  MATTER. 

the  suggestion  (though  it  has  never  yet  been  proven)  that  the 
attraction  existing  between  the  molecules  of  a  gas  is  due  to 
the  same  ultimate  cause  as  gravitational  attraction  (whatever 
that  cause  may  be),  and  that  the  so-called  Newtonian  law  of 
inverse  squares  is  only  an  approximate  expression  for  the 
relation  between  gravitation  and  distance,  which  is  sensibly 
accurate  only  when  the  distance,  D,  greatly  exceeds  the 
dimensions  of  a  molecule.  If  this  conjecture  be  correct,  the 
true  law  of  gravitational  attraction  is  probably  a  function 
capable  of  development  in  terms  of  descending  powers  of  D, 
thus  : 

f(V)  =  ^  +  ^  +  ^  +  -  (26) 

where  the  terms  after  the  first  have  very  small  coefficients,  so 
that  they  are  insensible  except  when  D  is  itself  extremely 
small,  and  the  reciprocals  of  its  higher  powers  correspondingly 
large. 

Equations  of  Van  der  Waals  and  Clausius. — The  equation 
pv  =  Rr  (27) 

being  deducible  from  the  kinetic  theory  only  when  the  as- 
sumption is  made  that  the  effects  of  the  attractive  forces 
between  the  molecules  are  insensible,  and  it  being  admitted 
that  even  in  the  so-called  permanent  gases  these  forces  are 
not  absolutely  insensible,  it  would  naturally  be  expected  that 
accurate  observations  would  show  that  this  equation  is  not 
perfectly  fulfilled  by  any  gas.  And  it  is  so.  We  can  no 
longer  regard  this  simple  law  as  anything  more  than  a  very 
good  first  approximation  to  the  truth.  All  gases  exhibit 
variations  from  it,  and  these  variations  are  quite  marked  at 
high  pressures,  when  the  molecules  are  crowded  closely 
together.  Many  formulae  have  been  proposed  as  "second 
approximations,"  and  of  these  the  most  famous  is  undoubtedly 
that  given  by  Van  der  Waals.  over  which  there  has  been 
much  controversy.  It  will  be  evident  to  you,  I  think,  that 


EQUATIONS    OF   VAN   DER   WAALS   AND   CLAUSIUS.    57 

the  "volume"  entering  into  our  expressions  should  not  be, 
strictly  speaking,  the  actual  space  occupied  by  the  gas,  but 
rather  the  space  in  which  the  molecules  are  free  to  move ; 
that  is,  the  empty  space  within  the  enclosure,  or  the  space  not 
actually  filled  by  the  molecules  themselves.  We  may  there- 
fore use  the  expression  (v  —  b)  in  the  place  of  v  in  the  fore- 
going formula,  v  still  signifying  the  apparent  volume  of  the 
gas,  while  (v  —  b)  is  the  empty  part  of  it.  The  gas  equation 
will  then  read 

(28) 


This  formula  is  an  improvement  on  the  older  one,  but  we 
have  not  yet  taken  account  of  the  attractive  forces  between 
the  molecules.  If  we  should  compress  a  gas,  we  should  be 
assisted  by  this  internal  attraction ;  and  therefore  the  actual 
external  pressure  required  to  reduce  the  volume  by  a  given 
amount  would  be  less  than  that  calculated  by  either  (27)  or 
(28).  This  much,  I  think,  is  admitted  by  all  physicists ;  but 
there  is  a  considerable  disagreement  among  them  as  to  the 
actual  algebraic  form  of  the  correction  thus  called  for.  Van 
der  Waals  considered  it  to  be  expressible  by  a  term  of  the 

form  —^ ;  and  his  equation  is,  therefore, 

•Kf          —  i       ,    *~  i  ,  -~         /29^ 


(v  —  b)      v* 
Clausius  proposed,  in  the  place  of  — 2 ,  the  expression 


so  that  his  equation  is  : 


However  interesting  Van  der  Waals's  equation  may  be,  it 
certainly  does  not  represent  the  facts  of  observation  satis- 


58       THE  MOLECULAR  THEORY  OF  MATTER. 

factorily,  unless  we  cease  to  regard  a  and  b  as  constants. 
Clausius's  equation,  on  the  other  hand,  is  found  to  represent 
the  results  of  Andre ws's  extensive  experiments  on  C02,  with 
great  fidelity.  The  values  of  the  constants,  in  (30),  for  C02 
are  as  follows  : 

R=    .003683,  b  =  .  000843, 

c  =  2.0935,  ft  =  .000977. 

The  pressures  are  to  be  reckoned  in  atmospheres,  the  temper- 
ature is  to  be  measured  on  the  absolute  Centigrade  scale,  and 
the  unit  of  volume  is  the  volume  of  the  gas  itself  at  the 
freezing-point,  and  at  atmospheric  pressure. 

Diffusion.  —  There  are  two  properties  of  gases  to  which  I 
must  refer  before  passing  to  the  consideration  of  liquids. 
The  first  of  these  is  diffusion.  It  has  often  been  remarked, 
by  persons  not  conversant  with  the  kinetic  theory,  that  if  the 
molecules  of  gases  are  moving  at  the  rate  of  thousands  of 
feet  per  second,  it  is  hard  to  understand  why  one  gas 
does  not  diffuse  through  another  with  correspondingly  great 
rapidity ;  so  that  if  a  bottle  of  some  strong-smelling  gas,  like 
ammonia,  were  opened  in  one  part  of  a  room  we  could  smell 
it  in  another  part  without  any  sensible  lapse  of  time,  even  if 
the  air  of  the  room  were,  apparently,  at  perfect  rest.  The 
answer  to  this  is,  that  although  the  molecules  of  ammonia 
vapor  do  have  high  velocities,  they  cannot  travel  in  uninter- 
rupted straight  lines  across  the  room,  like  projectiles,  because 
there  are  enormous  numbers  of  air  molecules  in  the  way, 
with  which  they  collide  thousands  of  millions  of  times  in  a 
second.  In  a  rough  way,  the  case  may  be  likened  to  an  army 
of  swift  runners  trying  to  pass,  at  the  top  of  their  speed, 
across  a  field  thickly  set  with  posts.  Of  course  this  is  a  very 
crude  comparison,  but  it  may  serve  to  fix  the  idea.  The 
runners  would  be  merely  turned  aside,  first  one  way  and  then 
another;  but  the  ammonia  molecules,  rebounding  from  the  air 
molecules  that  they  strike,  are  actually  turned  about,  and  in 


VISCOSITY.  59 

such  diverse  directions,  that  in  any  given  region  there  are 
almost  as  many  of  them  returning  towards  the  bottle  as  there 
are  going  away  from  it.  They  are  forced  to  describe  zig-zag 
paths  which  are  so  very  crooked  that  by  the  time  a  given 
ammonia  molecule  has  reached  a  point  actually  ten  feet  dis- 
tant from  the  bottle,  it  has  traveled,  in  all,  an  enormously 
greater  distance  —  probably  quite  a  number  of  miles.  I  shall 
not  give  you  the  mathematical  theory  of  diffusion,  because  it 
is  very  much  involved.  Furthermore,  I  do  not  consider  that 
any  very  valuable  information  has  yet  been  obtained  from 
it.  I  say  this  with  all  due  respect  to  the  men  who  have 
worked  over  this  theory  so  patiently.  What  they  have 
learned  about  diffusion  has  been  found  out  by  great  labor, 
and  it  is  gratifying  to  know  that  the  results  of  this  labor 
accord  well,  in  general,  with  similar  results  obtained  by  other 
methods.  There  is  no  doubt  that  very  important  facts  will 
be  learned,  some  day,  from  the  study  of  diffusion,  and  I 
should  not  want  to  discourage  any  of  you  who  may  feel 
disposed  to  enter  the  lists  and  give  battle  to  the  formulae 
involved. 

Viscosity.  — The  other  property  of  gases  to  which  I  referred 
a  moment  ago  is  viscosity.  It  is  by  no  means  an  obvious 
property,  and  its  effects  are  quite  small  —  so  small,  in  fact, 
that  they  are  difficult  to  measure  with  precision.  You  are 
probably  better  acquainted  with  the  viscosity  of  liquids  than 
you  are  with  the  corresponding  property  of  gases.  Molasses 
and  maple  syrup  are  familiar  examples  of  viscous  liquids. 
You  will  understand  that  if  two  planes  were  submerged  in 
molasses,  and  one  of  them  was  caused  to  move  in  its  own 
plane,  parallel  to  the  other  one  and  near  it,  there  would  be  a 
certain  drag  exerted  on  the  motionless  plane,  tending  to  pull 
it  in  the  direction  in  which  the  first  plane  is  moving.  This 
"  drag  "  would  also  be  experienced  by  the  fixed  plane  if  any 
other  liquid  were  substituted  for  the  molasses  ;  and  it  is 
found  that  the  same  effect  is  observable,  though  in  a  muck 


60      THE  MOLECULAR  THEORY  OF  MATTER. 

smaller  degree,  even  when  the  experiment  is  performed  with 
gases  instead  of  with  liquids.  This  property  of  fluids,  in 
virtue  of  which  they  can  transmit  motion  from  one  moving 
plane  to  another  one  parallel  to  it,  is  called  viscosity.  In 
many  respects  viscosity  is  analogous  to  friction;  and  physi- 
cists (particularly  among  the  Germans)  often  call  it  the 
"internal  friction"  of  the  fluid.  In  Fig.  21,  A  represents 
the  moving  plane,  and  B  the  fixed  one.  If  the  fluid  between 
them  be  conceived  to  be  divided  into  exceedingly  thin  layers, 
as  suggested  in  the  diagram,  we  may  conceive  the  resistance 


•-» 

1 

1 

1 

I 

I 

1                                         -p                       1 

1 

1 

] 

1 

1 

1 

1 

1 

B 

FIG.  21.— DIAGRAM  ILLUSTRATING  VISCOSITY. 

to  the  motion  of  A  to  be  due  to  the  friction  of  these  layers 
upon  one  another.  If  F  is  the  force  required  to  overcome 
this  friction  —  or,  which  is  the  same  thing,  if  F  is  the  force 
required  to  sustain  the  motion  of  A,  —  it  may  be  shown* 
that 

*-¥• 

where  S  is  the  area  of  the  plane  A,  V  is  its  velocity,  ,Z>  is 
the  perpendicular  distance  between  A  and  B,  and  ft  is  a 
coefficient  peculiar  to  the  fluid  experimented  with,  and  called 
its  coefficient  of  viscosity. 

*  See  Encyclopedia  Britannica,  article  Hydro-mechanics,  part  III. 


EXPERIMENTAL  DETERMINATION  OF  /a. 


61 


Experimental  Determination  of  /*.  —  The  foregoing  equa- 
tion may  be  written  in  the  form 


DF 

sv' 


(31) 


Hence  the  coefficient  of  viscosity  of  a  fluid  can  be  determined, 
if  we  can  measure  the 
quantities  D,  F,  S,  and 
V.  The  only  one  of  these 
that  presents  the  least 
difficulty  is  F,  which,  for 
gases,  is  so  small  that  its 
determination  is  ex- 
tremely difficult.  Various 
methods  have  been  tried, 
however,  and  owing  to 
the  skill  and  patience  of 
the  experimenters,  the 
results  are  fairly  satisfac- 
tory. Meyer  determined 
F  by  a  study  of  the  action 
of  air  on  vibrating  pen- 
dulums. He  worked  with 
three  pendulums,  and 
considered  that  the  most 
trustworthy  result  was 
that  obtained  from  the 
shortest  one  of  the  three. 
The  value  of  F  thus  de- 
duced by  him  gave 


FIG.  22.  —  DIAGRAM  ILLUSTRATING 
MAXWELL'S  APPARATUS. 


fi  =  .  000184, 

the  units  being  the  centi- 
meter, gramme,  and  second.      Another  method,  tried   with 
success  by  Maxwell  (and  Meyer  also),  consisted  in  causing 
a  circular  disk  to  oscillate  in  its  own  plane,  between  two 


62       THE  MOLECULAR  THEORY  OF  MATTER. 

others  that  were  parallel  to  it.  The  way  in  which  the  oscil- 
lations died  out  gave  a  means  of  calculating  F,  and  then,  by 
substitution  in  (31),  the  value  of  /x  was  obtained.  In  the 
actual  experiment,  Maxwell  used  several  disks  fixed  to  a  com- 
mon axis,  instead  of  only  a  single  one.  Fig.  22  is  a  diagram 
illustrating  the  principle  of  his  apparatus.*  Maxwell's  final 
result  for  the  coefficient  of  viscosity  of  dry  air  was 

P  =  .0001878  (1  +  .00365  1).  f 
Meyer's  result  was 

/*  =  .  000190(1  +  .00250- 

In  both  of  these  expressions  t  is  the  temperature,  on  the 
Centigrade  scale,  counted  from  the  freezing  point  of  water 
in  the  usual  way.  Maxwell's  expression,  as  quoted  above, 
implies  that  /*  varies  directly  as  the  absolute  temperature. 
Meyer  found,  by  experiment,  that  it  varies  in  proportion  to 
the  .77  power  of  the  absolute  temperature.  That  is,  it  varies 
proportionally  to 


where  £0  =  273°,  or  the  absolute  temperature  of  the  freezing 
point  on  the  Centigrade  scale.  The  best  value  of  the 
coefficient  of  viscosity  of  dry  air  at  present  obtainable  is 
considered  by  KnottJ  to  be 

fl  =  .  000185(1  +.0028^.  (32) 

Professor  Knott  also  gives  a  table  of  the  relative  values  of 
the  coefficients  of  viscosity  of  other  gases,  as  determined  by 
Graham,  Maxwell,  Meyer,  Kundt  and  Warburg,  and  Crookes. 

*  For  an  engraving  of  the  actual  apparatus,  see  the  Philosophical 
Transactions  of  the  Eoyal  Society  of  London,  for  1866. 

t  In  all  that  is  said  in  this  volume  about  viscosity,  the  fundamental 
units  are  understood  to  be  the  centimeter,  gramme,  and  second. 

\  Encyclopedia  Britannica,  article  Pneumatics. 


KINETIC   EXPLANATION   OF   VISCOSITY. 


63 


From  this  table  we  may  perhaps  infer  the  following  relative 
and  absolute  values  of  the  coefficients  of  viscosity  of  these 
gases,  at  the  freezing  point  of  water : 

TABLE  OF  COEFFICIENTS  OF  VISCOSITY  OF  GASES. 


(lAS 

COEFFICIENI 

c  OF  VISCOSITY. 

Relative. 

Absolute. 

Air 

1  000 

M  =  .000185 

Oxygen      • 
Nitrogen    

1.109 
.972 

.000205 
.000180 

Hydrogen 

.484 

.000090 

Carbonic  Oxide  .... 
Carbonic  Acid    .... 

.970 

.855 

.000179 
.000158 

Kinetic  Explanation  of  Viscosity.  —  Eeturning  to  our  con- 
ception of  a  gas  as  an  aggregation  of  swiftly  moving  mole- 
cules, let  us  see  if  it  is  not  adequate  to  explain  the  viscosity 
that  experiment  has  shown  these  bodies  to  possess.  Referring 
back  to  Fig.  21,  let  us  conceive  the  imaginary  layers  of  gas 
there  represented  to  be  of  such  a  thickness  that  the  average 
"free  path  "  of  the  molecules  of  the  gas  will  be  just  sufficient 
to  allow  these  molecules  to  cross  one  layer  in  the  interval 
between  two  successive  collisions.  Now  when  the  molecules 
in  the  upper  layer  strike  against  A,  they  receive  from  it  a 
component  motion  in  the  direction  of  the  arrow.  Rebound- 
ing from  Ay  they  cross  the  first  layer  and  collide  with  the 
molecules  of  the  second  layer,  communicating  to  them  also 
a  component  motion  in  the  direction  of  the  arrow.  The 
molecules  in  the  second  layer  carry  this  component  over  to 
those  in  the  third  layer,  and  so  on  until  the  molecules  in 
the  last  layer  tend  to  communicate  a  component  motion, 
in  the  direction  of  the  arrow,  to  the  plane  B.  This  is  the 
general  nature  of  the  kinetic  explanation  of  viscosity  in  gases, 
though  you  will  understand  that  in  the  rigorous  mathematical 


64       THE  MOLECULAR  THEORY  OF  MATTER. 

treatment  of  the  subject  it  is  necessary  to  take  account  of 
many  things  that  I  have  not  mentioned;  such,  for  instance, 
as  the  law  of  distribution  of  velocities  among  the  molecules. 
Several  mathematicians,  following  out  the  general  idea  I  have 
given  you,  have  deduced  from  the  kinetic  theory  of  gases, 
expressions  for  the  coefficient  of  viscosity.  One  of  the  most 
recent  of  these  expressions  is  that  of  Clausius,  which  is 

(33) 

> 

where  A  is  the  density  of  the  gas  at  0°  C.  and  atmospheric 
pressure,  as  compared  with  air  under  the  same  conditions, 
T  is  the  absolute  temperature  of  the  gas,  T0  is  the  absolute 
temperature  of  the  freezing  point  of  water  (273°  C.),  and  X 
is  the  average  "free  path"  of  the  molecules  of  the  gas,  at 
atmospheric  pressure  and  at  0°  C.  While  it  is  true  that  this 
formula  does  not  accurately  represent  the  experimental  facts, 
inasmuch  as  the  exponent  of  the  absolute  temperature  is 
known  to  be  about  .77  instead  of  -j-,  the  error  introduced  in 
this  way  is  practically  insensible  for  temperatures  near  the 
freezing  point.*  There  is  one  very  important  thing  I  would 
like  you  to  notice  about  this  formula.  It  is,  that  the  formula 
indicates  that  the  coefficient  of  viscosity  of  a  gas  remains 
unchanged,  no  matter  how  much  we  rarefy  the  gas  or  compress 
it;  for  X  and  A  are  constants,  since  they  stand  for  the  free 
path  and  density  at  atmospheric  pressure  and  at  0°  C.  That 

*  Mr.  Sutherland  has  shown  very  clearly  that  the  difference  between 
theory  and  experiment,  here  mentioned,  disappears  if  we  take  account 
of  the  attractive  forces  existing  between  the  molecules  of  the  gas.  Assum- 
ing that  these  forces  are  proportional  to  the  inverse  fourth-power  of  the 
distance,  he  finds  that 


Mo 


where  MO  is  the  coefficient  of  viscosity  at  the  freezing  point  and  M  is  the 
corresponding  coefficient  at  the  absolute  temperature  T.     The  value  of 


.  KINETIC    EXPLANATION    OF 

the  viscosity  of  a  gas  does  not  depend 
density,  lias  been  amply  verified  by  experiment.  Numerous 
observers  have  found  that  air  at  a  pressure  of  only  a  few 
millimeters  of  mercury  has  substantially  the  same  viscosity 
as  air  under  the  normal  pressure  of  760  millimeters.  It  is 
hardly  necessary  to  say  that  this  constitutes  a  most  remark- 
able verification  of  the  kinetic  theory.  Of  course  this  state- 
ment about  viscosity  being  independent  of  absolute  density 
must  not  be  pressed  to  extremes,  or  we  should  be  saying  that 
a  vacuum  has  just  as  much  viscosity  as  common  air ;  which 
would  be  absurd.  A  vacuum  cannot  have  viscosity.  This 
does  not  involve  an  error  in  Clausius's  reasoning,  however, 
for,  as  he  says  himself,  his  formula  was  deduced  on  the 
assumption  that  the  free  path  is  so  small  that  terms  involv- 
ing higher  powers  of  it  than  the  first  can  be  neglected. 
There  is  no  doubt  about  this  being  the  fact  under  ordinary 
conditions ;  but  when  the  gas  is  rarefied  the  molecules  have 
much  more  free  space  to  move  in.  They  do  not  collide  so 
often,  and  hence  their  average  free  path  becomes  greater. 
As  the  exhaustion  proceeds,  there  comes  a  time  when  the 
terms  involving  powers  of  the  average  free  path  higher  than 
the  first  cannot  be  neglected ;  and  when  that  point  is  reached, 
the  formula  no  longer  has  any  pretensions  to  accuracy. 

the  constant  C,  for  C02,  is  277.     The  following  table  exhibits  the  agree- 
ment of  this  formula  with  Holman's  experiments  on  COa  : 


TABLE  OF  VALUES  OF  THE  RATIO  —  FOR  C02. 

Mo 


TEMP. 

OBSERVED 
RATIO. 

CALCULATED 
KATIO. 

TEMP. 

OBSERVED 
KATIO. 

CALCULATED 
KATIO. 

18°  C. 

1.068 

1.066 

119°.4  C. 

1.415 

1.414 

41° 

1.146 

1.148 

142° 

1.484 

1.490 

59° 

1.213 

1.211 

158° 

1.537 

1.541 

79°.  5 

1.285 

1.280 

181° 

1.619 

1.614 

100°.2 

1.351 

1.351 

224° 

1.747 

1.746 

66 


THE  MOLECULAR  THEORY  OF  MATTER. 


Free  Path.  —  Clausius's  formula  is  of  no  particular  value 
for  computing  the  coefficient  of  viscosity  of  gases,  because  it 
involves  the  free  path,  A,  which  is  much  harder  to  determine 
than  the  viscosity  is ;  but  when  the  coefficient  of  viscosity 
has  once  been  determined  by  experiment,  Clausius's  formula 
becomes  of  the  most  extreme  importance  for  the  inverse 
problem  of  determining  the  average  free  path.  Thus,  con- 
fining our  attention  to  the  case  in  which  T  =  T0,  we  may 
write  the  formula  in  the  form 


20.23 


(34) 


Applying   this    equation    to    carbonic   acid   gas,   for   which 
A  =  1.529  and  /A  =  .000158,  we  have 

.000158 


20.23 


=  .000  0063  cm. 


Similar  calculations  for  the  other  gases  whose  coefficients  of 
viscosity  are  in  the  table  I  gave  you  a  few  moments  ago, 
give  the  following  results  : 

TABLE  OF  AVERAGE  FREE  PATHS  AT  ATMOSPHERIC  PRESSURE  AND  0°  C. 


DENSITY. 

AVERAGE  I 

REE  PATH. 

(Air  =  l.) 

Centimeters. 

Inches. 

Air*                      .     .     . 

1.0000 

.000  00914 

.000  0036 

Oxv°"en 

1  1056 

.000  00964 

.000  0038 

Nitrogen 

9713 

.000  00903 

.000  0036 

Hydrogen    

.06926 

.000  01690 

.000  0067 

Carbonic  Oxide    .... 
Carbonic  Acid      .... 

.9545 
1.5290 

.000  00906 
.000  00631 

.000  0036 
.000  0025 

I  think  these  figures  will  make  it  plain  why  one  gas  does 
not  diffuse  through  another  with  a  speed  comparable  with  the 
velocity  of  translation  of  its  molecules.  Consider  the  case  of 

*  Calculated  as  a  simple  gas. 


FREE    PATH. 


67 


hydrogen,  for  example.  We  found  the  average  velocity  of 
translation  of  its  molecules  to  be  5,571  feet  per  second ;  and 
as  the  average  free  path  of  its  molecules,  between  collisions, 
is  .000  0067  of  an  inch,  you  will  see  that,  on  an  average,  each 
of  its  molecules  experiences 

:  ^#=10,000,000,000* 

collisions  with  its  neighbors  every  second.  When  one  gas 
is  diffusing  through  another  one,  the  calculation  is  not  so 
simple ;  because  in  that  case  there  are  two  kinds  of  molecules 
to  be  considered,  and  we  must  take  account  of  the  collisions 
of  each  molecule  with  those  of  its  own  set,  and  with  those  of 
the  other  set.  Still,  I  think  you  will  be  quite  ready  to  admit, 
from  the  calculation  I  have  given  you  in  the  case  of  hydrogen, 
that  the  number  of  collisions  per  second  experienced  by  each 
of  the  molecules  of  a  diffusing  gas  would  be  so  great  that  the 
explanation  of  the  slowness  of  diffusion  that  I  offered  you  a 
few  moments  ago  is  quite  defensible.  It  may  interest  you  to 
know  the  average  number  of  collisions  per  second  experienced 
by  a  molecule  in  some  other  gases.  I  have  therefore  cal- 
culated this  table,  by  the  same  method  used  in  the  case  of 
hydrogen,  and  from  the  data  concerning  the  free  paths  and  the 
molecular  velocities  that  have  already  been  placed  before  you. 

TABLE  OF  THE  AVERAGE  NUMBER  or  COLLISIONS  PER  SECOND  EXPE- 
RIENCED BY  A  MOLECULE  OF  VARIOUS  GASES,  AT  ATMOSPHERIC 
PRESSURE  AND  0°  C. 


GAS. 

COLLISIONS  PER  SECOND. 

Oxygen 

4,410  000,000 

Nitrogen  . 

5  021  000  000 

Hydrogen 

10  040  000  000 

Carbonic  Oxide  .     .     .     . 
Carbonic  Acid  .... 

5,014,000,000 
5,741,000,000 

*  In  round  numbers. 


68       THE  MOLECULAR  THEORY  OF  MATTER. 

High  Vacua.  —  The  average  free  path  of  the  molecules  of  a 
given  gas  is  independent  of  the  temperature,  but  it  varies 
when  the  density  of  the  gas  is  made  to  vary.  For  the  average 
distance  a  molecule  will  travel,  between  successive  collisions, 
will  obviously  be  less  when  there  are  many  molecules  in  a 
unit  volume  than  it  will  be  when  there  are  but  few  of  them. 
In  fact,  mathematical  analysis  shows  that  the  length  of  the 
average  free  path  is  exactly  proportional  to  the  reciprocal  of 
the  density  of  the  gas.  It  follows  from  this  that  we  can 
make  the  average  free  path  as  great  as  we  please,  by  dimin- 
ishing the  density  of  the  gas  sufficiently.  If,  therefore,  we 
should  diminish  the  density  by  the  aid  of  a  good  modern  air 
pump  until  it  were  only  (say)  a  millionth  of  its  normal  value 
at  atmospheric  pressure,  we  should  thereby  increase  the 
average  free  path  of  the  molecules  to  one  million  times  the 
lengths  I  have  given  you  in  the  table.*  Thus  in  the  case 
of  C02  the  average  free  path  would  become  about  two  inches 
and  a  half,  and  in  the  case  of  hydrogen  it  would  even  become 
six  inches  and  three-quarters.  It  would  be  natural  to  think 
that  at  such  high  exhaustions  the  residual  gas  would  exhibit 
no  phenomena  at  all  —  that  it  would  be  indistinguishable  from 
a  vacuum.  But  this  is  not  the  case.  We  must  remember  that 
the  contents  of  a  vessel  exhausted  to  this  degree  is  called  a 
vacuum  "by  courtesy  only."  There  are  still  many  millions 
of  molecules  left  in  it ;  and  as  their  mean  free  paths  are  now 
measurable  in  inches,  the  medium  exhibits  entirely  new  prop- 
erties, some  of  which  I  shall  try  to  show  you. 

The  Radiometer.  —  I  have  here  a  small  piece  of  appara- 
tus (Fig.  23),  kindly  loaned  for  the  evening  by  Professor 
Kimball,  from  the  physical  laboratory  of  the  Institute.  You 
will  see  that  it  consists  of  a  glass  bulb  in  which  is  a  sort  of 
wind-mill,  mounted  very  delicately  upon  a  pivot.  The  vanes 
of  this  wind-mill  are  blackened  on  one  side,  while  on  the  other 
side  they  are  bright.  I  will  hold  the  bulb  near  the  gas  jet. 
*  On  page  66. 


THE    RADIOMETER. 


69 


You  see  that  the  vanes  are  now  flying  around  very  rapidly, 
bright  side  foremost.  I  hold  it  still  nearer  the  gas  jet,  and 
the  vanes  revolve  with  such  speed  that  they  cannot  be  sep- 
arately distinguished.  This  wonderful  little  instrument  is 
called  the  radiometer;  and  I  am  going  to  try  to  make  it  clear 
to  you  why  the  vanes  revolve.  The 
ultimate  phenomena  on  which  the 
motion  depends  are  somewhat  com- 
plicated, and  I  shall  only  attempt  to 
give  you  a  general  idea  of  them.* 
In  the  first  place,  the  blackened  sides 
of  the  vanes  absorb  more  of  the 
radiant  energy  from  the  gas  jet  than 
the  bright  sides  do,  and  hence  they 
become  warmer.  This  difference  in 
temperature  is  essential  to  the  work- 
ing of  the  instrument ;  and  hence  it 
is  important  to  make  the  vanes  of 
some  fairly  good  non-conductor,  such 
as  mica,  in  order  that  the  tempera- 
tures of  their  opposite  surfaces  may 
not  become  equalized  by  conduction. 
The  difference  in  temperature  be- 
tween the  black  and  bright  surfaces 
is  probably  small,  but  for  brevity  we 
may  speak  of  these  surfaces  as  the 
"hot  side"  and  the  "cold  side," 
respectively.  The  molecules  on  the 
hot  side  of  a  vane  are  vibrating 

more  energetically  than  they  are  on  the  cold  side  and  hence 
they  communicate  heavier  blows  to  such  gas-molecules  as 
chance  to  collide  with  them.  Now,  since  action  and 
reaction  are  equal,  it  follows  that  the  gas-molecules  react 
more  powerfully  on  the  hot  (or  black)  surfaces  than  they  do 

*  For  a  more  satisfactory  discussion,  see  Maxwell's  article  in  the  Phil- 
osophical Transactions  for  1879. 


FIG.  23.  — THE  RADIOMETER. 


70       THE  MOLECULAR  THEORY  OF  MATTER. 

on  the  cold  ones,  and  hence  there  is  a  tendency  to  drive  the 
vanes  around  as  you  saw  them  go,  bright  side  foremost.  This 
tendency  is  not  sufficient  to  cause  the  vanes  to  revolve  when 
the  bulb  is  filled  with  air  of  ordinary  density,  however, 
because  the  average  free  path  of  the  air-molecules  is  so 
extremely  small,  and  the  number  of  molecular  collisions  per 
second  so  enormously  great,  that  the  accelerated  molecules 
that  fly  off  from  the  hot  sides  of  the  vanes  beat  back  the 
molecules  of  air  in  the  immediate  neighborhood  of  the  vane 
sufficiently  to  cause  a  slight  rarefaction  of  the  air  in  front 
of  the  hot  side.  This  rarefaction  tends  to  make  the  vanes 
revolve  black  side  foremost.  When  the  air  in  the  bulb  is 
of  ordinary  density,  these  two  opposing  tendencies  appear  to 
be  sensibly  balanced,  and  no  motion  results.  When  the  air  in 
the  bulb  is  moderately  exhausted,  the  effect  due  to  local  rare- 
faction in  front  of  the  hot  side  preponderates,  and  the  vanes 
slowly  revolve  black  side  foremost;  but  when  the  exhaustion 
is  pushed  to  such  a  degree  that  the  free  path  of  the  residual 
molecules  becomes  as  great  as  the  distance  from  the  vanes 
to  the  glass  bulb,  the  accelerated  molecules  no  longer  beat 
back  their  neighbors  ;  there  is  no  local  rarefaction ;  the  only 
remaining  cause  of  motion  is  the  reaction  of  the  black  sur- 
faces against  the  accelerated  gas-molecules;  and  the  vanes 
therefore  revolve  bright  side  foremost. 

Crookes's  Tubes.  —  The  radiometer  depends  for  its  action 
on  the  presence  of  the  walls  of  the  glass  bulb,  and  the  larger 
the  instrument  is,  the  more  perfect  must  the  exhaustion  of 
the  bulb  be ;  for  the  essential  thing  about  the  instrument  is, 
as  I  have  explained,  that  the  average  free  path  of  the  mole- 
cules of  the  residual  gas  must  be  great  enough  to  permit 
these  molecules  to  strike  against  the  walls  of  the  bulb,  after 
rebounding  from  the  vanes,  instead  of  against  one  another. 
I  have  here  three  other  pieces  of  apparatus,  the  operation  of 
which  does  not  depend  upon  the  dimensions  of  the  containing 
tube.  I  am  enabled  to  show  you  these  tubes  through  the 


CKOOKES'S   TUBES. 


71 


courtesy  of  Messrs.  Queen  &  Co.,  of  Philadelphia,  to  whom 
they  belong,  and  who  have  very  kindly  loaned  them  for  this 
occasion.  In  each  of  these  tubes  the  degree  of  exhaustion 
is  so  great  that  the  average  free  path  of  the  residual  mole- 
cules is  several  inches  long.  I  will  first  show  you  this  tube 
(Fig.  24),  in  which  the  vanes  of  the  little  fly  are  set  obliquely, 
and  are  not  blackened  at  all.  As  I 
connect  the  terminals  of  the  battery 
with  the  ends  of  the  loop  of  wire 
below  the  fly,  you  notice  that  this 
wire  becomes  hot.  The  gas-molecules 
that  collide  with  the  hot  wire  are 
driven  off  in  all  directions,  at  greatly 
increased  speed,  by  the  vigorous 
blows  they  receive  from  the  mole- 
cules of  the  wire,  which  are  now 
vibrating  very  energetically.  Those 
that  strike  against  the  vanes  of  the 
fly  impinge  on  them  obliquely  and 
cause  them  to  revolve,  as  you  see 
they  are  now  doing.  I  will  next 
connect  the  loop  and  the  upper  elec- 
trode with  the  respective  terminals 
of  the  induction  coil.  You  see  that 
this  also  causes  the  fly  to  rotate. 
Before  explaining  why  the  induction 
coil  causes  the  vanes  to  move,  I  will 
show  you  another  tube  (Fig.  25), 
which  is  very  simple  in  construction, 
but  very  beautiful  and  instructive 
in  operation.*  It  consists  of  two  electrodes  sealed  into 
the  tube,  one  of  which  is  concave,  or  cup-shaped.  As  I 
throw  the  coil  into  action,  you  notice  the  hazy  double  cone 
of  purplish  light,  the  vertex  of  which  is  at  the  center  of  cur- 

*  In  exhibiting  this  tube  and  the  next  one,  the  lights  in  the  lecture 
room  were  turned  down. 


FIG.  24.  — A  CROOKES'S  TUBE 
WITH  OBLIQUE  VANES. 


72       THE  MOLECULAR  THEORY  OF  MATTER. 

vature  of  the  concave  electrode ;  and  where  the  cone  spreads 
out  so  as  to  intersect  the  outer  tube,  the  glass  shines  with 
a  beautiful  golden  fluorescence.  Now  the  explanation  of  this 
phenomenon  is,  that  as  the  molecules  of  residual  gas  come  in 
contact  with  the  concave  electrode  they  receive  a  charge  of 
electricity  themselves,  and  are  energetically  repelled  in  a 
direction  normal  to  the  surface  of  the  electrode.  Flying 
away  from  the  electrode,  they  necessarily  pass  close  to  its 
center  of  curvature,  and  being  crowded  together  at  that  point, 
they  brush  against  one  another  sufficiently  to  give  rise,  in 
some  manner,  to  the  purplish  glow  that  you  see.  After  they 


FIG.  25.  —  A  CROOKES'S  TUBE  WITH  CONCAVE  ELECTBODE. 

have  passed  the  vertex  of  the  cone  of  light,  they  diverge  once 
more ;  and  when,  continuing  to  move  in  straight  lines,  they 
come  in  collision  with  the  sides  of  the  tube,  they  excite  a 
fluorescence  in  the  glass  which  dies  away  towards  the  remote 
end  of  the  tube,  as  the  flying  stream  of  molecules  strikes 
more  and  more  obliquely.  I  think  you  will  see,  now,  why 
the  induction  coil  caused  the  vanes  of  the  other  tube  (Fig.  24) 
to  revolve.  The  gas-molecules  were  projected  from  the  elec- 
trode by  electrical  repulsion,  and  the  effect  was  the  same 
as  when  the  wire  loop  was  directly  heated  by  the  battery. 
There  is  one  very  strange  thing  about  these  high  vacuum 
tubes,  which  I  think  no  one  has  yet  satisfactorily  explained. 


CROOKES'S   TUBES.  73 

You  will  notice  that  when  I  reverse  the  direction  of  the  cur- 
rent through  the  coil,  the  appearance  of  the  tube  (Fig.  25)  is 
entirely  changed.  There  is  no  longer  any  sign  of  the  double 
cone  of  flying  molecules.  It  appears  that  in  the  phenomena 
of  electric  repulsion  in  these  tubes,  it  is  the  negative  electrode 
exclusively  that  is  concerned.  It  seems  to  make  little  differ- 
ence what  part  of  the  tube  the  positive  electrode  is  in.  I  will 
now  show  you  the  remaining  tube  (Fig.  26),  and  I  think  you 
will  agree  with  me  that  it  is  exceedingly  beautiful  when  in 
operation.  It  consists  of  a  horizontal  tube  containing  a  pair 
of  parallel  glass  rails,  along  which  a  sort  of  little  paddle- 
wheel  can  roll.  The  electrodes,  you  will  notice,  are  on  a 
level  with  the  uppermost  vanes  of  the  wheel.  I  will  now 


FIG.  26.  — A  CROOKES'S  TUBE  WITH  ROLLING  WHEEL. 

start  the  coil  in  action,  and  you  see  the  whole  tube  glorious 
with  light  and  color.  The  molecular  stream  from  the  nega- 
tive electrode,  beating  on  the  upper  vanes,  causes  the  wheel  to 
revolve  and  roll  along  the  track  toward  the  other  end  of  the 
tube ;  but  just  before  it  gets  there  I  reverse  the  coil,  and  you 
see  the  wheel  come  to  a  stand-still  and  then  begin  to  revolve 
in  the  opposite  direction  until  it  returns  to  its  starting  point. 
We  can  make  it  travel  back  and  forth  as  many  times  as  we 
please,  by  merely  reversing  the  coil  when  the  wheel  nears  the 
end  of  its  course.  You  will  notice,  at  the  ends  of  the  tube 
and  along  the  bottom  of  it,  what  appears  to  be  the  shadow  of 
the  glass  rails.  It  is  not  a  true  shadow,  however.  The 
gorgeous  fluorescence  that  you  see  elsewhere  is  caused,  as  in 
the  last  tube  I  showed  you,  by  the  impact  of  the  molecules 


74       THE  MOLECULAR  THEORY  OF  MATTER. 

against  the  sides  of  the  tube;  and  the  dark  places  do  not 
shine,  simply  because  they  are  shielded  from  the  molecules 
that  are  streaming  away  in  straight  lines  from  the  negative 
electrode.  To  prove  this  to  you,  I  move  this  small  magnet 
about  in  the  neighborhood  of  the  tube,  and  you  see  the  dark 
lines  shifting  from  place  to  place,  as  they  would  not  do  if 
they  were  true  shadows.  The  streams  of  electrified  mole- 
cules, behaving  in  a  certain  sense  like  electric  currents,  are 
deflected  by  the  magnet ;  and  different  parts  of  the  tube  are 
shielded  as  I  change  the  magnet  into  different  positions,  so 
that  the  pseudo-shadows  appear  to  move  about.  Before  leav- 
ing this  interesting  subject,  let  me  say  that  these  wonderful 
mechanical  phenomena  in  high-vacuum  tubes  were  discovered 
by  Mr.  William  Crookes,  whose  researches  in  this  department 
of  physics  have  earned  him  a  lasting  renown.  He  has  given 
us  an  experimental  demonstration  of  the  kinetic  theory  of 
gases,  and  in  his  apparatus  we  can  almost  see  the  molecules 
as  they  fly  about. 

III.    THE  MOLECULAR  THEORY  OF  LIQUIDS. 

Preliminary  Remarks.  —  I  have  already  given  you  the 
kinetic  definition  of  a  liquid.*  You  will  remember  that 
liquids  resemble  gases  in  one  respect,  which  is,  that  their 
molecules  can  move  freely  about  among  one  another.  They 
differ  from  gases,  however,  in  having  a  much  smaller  average 
molecular  velocity,  and  in  having  their  molecules  so  close 
together  that  they  are  always  well  within  the  sphere  of  one 
another's  attractive  influence.  Except  when  great  accuracy 
is  required,  we  found  it  possible  to  ignore  the  intermolecular 
forces  in  gases.  This  makes  the  molecular  theory  of  these 
bodies  comparatively  simple.  In  liquids,  however,  no  such 
simplification  is  possible,  for  the  molecular  forces  are  no 
longer  insensible.  Their  effects  are  everywhere  visible,  and 
in  discussing  liquids  we  can  never  cease  to  consider  them. 

*  Page  12. 


FREE   EVAPORATION.  75 

This  fact,  and  the  more  lamentable  one  that  we  do  not  yet 
certainly  know  the  form  of  the  molecular  force-function  (that 
is,  the  law  of  variation  of  the  force  with  distance),  render 
the  study  of  liquids  exceedingly  difficult.  Thus  it  is  that 
although  questions  of  the  greatest  moment  are  arising  con- 
tinually in  this  field  of  molecular  physics,  to  most  of  them 
we  can  make  no  answer  at  present.  No  mathematician  has 
yet  worked  out  the  kinetic  theory  of  liquids  to  anything  like 
the  extent  to  which  the  corresponding  theory  of  gases  has 
been  pushed;  and  for  this  reason  what  I  shall  say  about 
liquids  will  necessarily  be  of  a  fragmentary  character. 

Free  Evaporation.  —  There  can  be  no  doubt  that  the  mole- 
cules composing  liquids  have  as  great  a  variety  of  velocities 
as  those  in  gases ;  for  there  must  be  almost  innumerable 
collisions  among  them,  and  even  if  there  were  an  absolute 
equality  of  velocities  at  any  given  instant,  the  collisions 
would  necessarily  destroy  this  equality  at  once.  Doubtless 
there  is  some  law  of  distribution  of  velocities  in  liquids, 
corresponding  to  Maxwell's  law  in  gases ;  *  but  the  form  of 
this  law  has  not  yet  been  discovered.  Admitting  the  fact 
that  the  velocities  of  the  molecules  are  unequal,  let  us  con- 
sider what  would  happen  at  a  free  surface  of  the  liquid, 
assuming  for  the  moment  that  above  this  free  surface  there 
is  a  boundless  vacuum.  A  particle  well  within  the  liquid  is 
attracted,  on  the  whole,  equally  in  all  directions.  A  particle 
at  the  surface,  however,  is  attracted  only  dowmvard.  You 
will  see,  therefore,  that  when  a  molecule,  in  the  course  of 
its  wanderings,  comes  to  the  surface,  whether  it  will  escape 
from  the  liquid  or  not  depends  upon  the  magnitude  of  the 
vertical  component  of  its  motion.  If  this  vertical  component 
is  sufficient  to  carry  the  molecule  beyond  the  range  of  sensible 
attraction  of  the  liquid,  the  molecule  will  pass  away  indefi- 
nitely into  the  space  above.  On  the  other  hand,  if  the  vertical 
component  of  its  motion  is  not  sufficient  to  carry  the  mole- 

*  See  equation  (1). 


76       THE  MOLECULAK  THEORY  OF  MATTER. 

3 

cule  beyond  the  range  of  sensible  attraction  of  the  liquid, 
it  will  rise  into  the  vacuous  space  only  a  short  distance,  its 
upward  velocity  growing  less  and  less  under  the  influence  of 
the  downward  attractive  forces  until  it  vanishes  altogether, 
after  which  the  molecule  will  begin  to  fall  back  again,  and  it 
will  finally  plunge  once  more  into  the  liquid.  At  the  surface 
of  a  liquid,  therefore,  the  molecules  are  continually  describ- 
ing paths  something  like  those  indicated  in  this  diagram 
(Fig.  27),  where  the  dotted  line  represents  the  limit  of  sen- 


FIG.  27. —  DIAGRAM  OF  A  LIQUID  SURFACE. 

sible  molecular  attraction.  Most  of  the  molecules  that  start 
upward,  fall  back  into  the  liquid ;  and  the  escape  of  such  of 
them  as  are  moving  fast  enough  to  overcome  the  attraction  of 
the  liquid,  constitutes  the  phenomenon  that  we  call  free  evap- 
oration. I  said  that  of  those  molecules  that  leave  the  liquid, 
the  majority  fall  back  again ;  and  perhaps  I  ought  to  explain 
how  we  know  this  to  be  the  fact.  If  the  converse  were  true 

—  that  is,  if  the  majority  of  them  at  once  flew  off  into  space 

—  we  should  have  to  conclude  that  the  average  molecular 
velocity  in  liquids  is  just  about  great  enough  to  overcome  the 
attraction  of  the  liquid  for  a  molecule  about  to  leave  it ;  and 
the  enormous  latent  heat  of  vaporization  of  liquids  proves 
that  this  cannot  be  so.     For  example,  we  have  to  add  a  vast 
amount  of  energy  to  a  pound  of  water  before  it  will  pass  into 
steam  ;  and  this  shows  that  under  ordinary  circumstances  the 


COOLING    EFFECT    OF    EVAPORATION.  77 

average  kinetic  energy  of  translation  of  a  water-molecule  is 
far  too  small  to  overcome  the  molecular  attractive  forces. 
Hence  it  follows  that  of  the  many  molecules  that  come  to 
the  surface  of  a  liquid  in  a  given  time,  very  few  will  per- 
manently escape,  because  very  few  have  velocities  sufficiently 
in  excess  of  the  average  to  enable  them  to  pass  away,  directly 
against  the  attractive  force  of  the  liquid. 

Cooling  Effect  of  Evaporation.  —  We  know  that  the  tem- 
perature of  a  gas  is  proportional  to  the  average  kinetic  energy 
of  translation  of  the  molecules  of  the  gas.  It  is  not  so  certain 
that  this  is  the  case  with  liquids,  for  these  bodies  are  consti- 
tuted so  differently  that  we  can  hardly  assume  them  to  act 
on  oiir  senses  in  precisely  the  same  manner.  Nevertheless, 
it  is  probable  that  there  is  some  analogous  relation  between 
the  kinetic  energy  of  a  liquid  and  the  temperature  of  the 
liquid ;  so  that  although  the  two  may  not  be  strictly  propor- 
tional, we  may  fairly  assume,  I  think,  that-  one  of  them  is 
what  mathematicians  would  call  a  continuous,  one-valued, 
increasing  function  of  the  other.  This  being  admitted,  it  is 
easy  to  see  why  evaporation  cools  a  liquid.  For  it  is  plain 
that  when  a  liquid  is  evaporating,  it  loses  only  those  mole- 
cules which  have  a  speed  considerably  greater  than  the 
average,  —  the  slower  moving  ones,  as  I  have  explained, 
being  retained  by  the  attraction  of  the  liquid.  This  is 
equivalent  to  saying  that  the  molecules  that  do  fly  off  will 
carry  away  with  them  more  than  their  equable  share  of  the 
kinetic  energy  of  the  liquid.  Hence  while  evaporation  is 
going  on,  the  average  kinetic  energy  per  molecule,  in  the- 
mother  liquid,  is  continually  growing  less ;  and  this  means; 
that  the  temperature  of  the  liquid  is  falling. 

Vapor  Density.  —  Thus  far  we  have  spoken  only  of  the 
phenomena  of  evaporation  in  a  boundless  vacuum,  and  we 
now  come  to  the  consideration  of  evaporation  in  a  closed 
vessel.  We  will  suppose  that  at  the  outset  this  vessel  is 


78       THE  MOLECULAR  THEORY  OF  MATTER. 

absolutely  empty,  and  that  at  a  certain  instant  a  small 
quantity  of  water  is  admitted  to  it.  During  the  first  instant 
following  the  introduction  of  the  water,  the  phenomena  are 
precisely  the  same  as  we  have  seen  them  to  be  in  the  case 
of  the  boundless  vacuum.  The  swiftest  molecules  fly  off  as 
before ;  but  they  can  no  longer  pass  away  indefinitely  into 
space.  They  are  now  retained  by  the  vessel,  in  which  they 
will  accumulate,  constituting  a  gas  or  vapor  whose  density 
will  go  on  increasing  until  a  certain  limit  is  reached.  You 
will  readily  see  that  the  molecules  composing  this  vapor  will 
travel  in  every  direction,  precisely  as  they  do  in  other  gaseous 
bodies.  Many  of  them,  therefore,  will  'plunge  back  into  the 
liquid  again,  and  become  an  integral  part  of  it  once  more. 
Now  the  number  of  molecules  that  leave  the  mother  liquid  in 
a  given  time  will  be  quite  independent  of  the  density  of  the 
vapor  overhead ;  but  the  number  that  fly  back  into  it  again, 
in  a  given  time,  will  be  greater,  the  greater  the  density  of  the 
vapor.  At  the  beginning  of  the  evaporation  the  vapor  will 
be  rare,  and  the  number  of  molecules  that  fly  off  in  any  given 
time  will  greatly  exceed  the  number  that  return  during  that 
time.  The  density  of  the  vapor  will  therefore  increase. 
After  a  certain  interval  (an  exceedingly  short  interval,  meas- 
ured by  ordinary  standards),  the  density  of  the  vapor  will 
become  so  great  that  tftfe  number  of  molecules  plunging  back 
into  the  liquid  in  a  given  time  will  become  sensibly  equal  to 
the  number  that  fly  off  from  it  in  the  same  time.  When  this 
adjustment  becomes  perfect,  the  density  of  the  vapor  will  no 
longer  increase.  It  is  then  said  to  be  "saturated."  I  would 
like  to  fix  it  clearly  in  your  minds  that  a  saturated  vapor  is 
one  in  which  the  number  of  molecules  that  plunge  back  into 
the  mother  liquid  in  any  given  time,  is  precisely  equal  to  the 
number  of  molecules  that  rise  out  of  the  liquid,  in  the  same 
time,  and  enter  the  vapor.  You  will  see  that  any  cause  that 
tends  to  disturb  this  equality  will  also  tend  to  alter  the 
density  of  the  vapor.  For  example,  if  we  raise  the  tempera- 
ture of  the  system  we  shall  destroy  the  equality  in  question ; 


VAPOR   PRESSURE.  79 

for  we  shall  accelerate  all  the  molecules,  and  hence  more 
molecules  will  plunge  from  the  vapor  into  the  liquid  in  a 
given  time  than  before,  and  more  molecules  will  also  come 
to  the  surface  of  the  liquid  from  the  interior.  Furthermore, 
of  the  increased  number  of  molecules  that  emerge  from  the 
interior  of  the  liquid,  a  larger  proportion  than  before  will 
have  velocities  exceeding  the  critical  velocity  that  a  molecule 
must  have  in  order  to  escape  from  the  attraction  of  its  fellows. 
Hence,  on  the  whole,  the  density  of  the  vapor  will  increase, 
approaching  a  new  limit  at  which  the  number  of  in-coming 
and  out-going  molecules  will  again  become  equal.  We  see, 
therefore,  that  for  any  given  vapor  in  contact  with  its  liquid 
there  is  a  definite  density  corresponding  to  each  temperature. 
If  we  knew  enough  about  the  physics  of  liquids  and  vapors, 
we  could  express  the  relation  between  temperature  and  vapor 
density  by  means  of  a  rational  equation ;  but  unfortunately 
we  are  still  very  far  indeed  from  possessing  this  knowledge. 
Our  reasoning  shows  that  the  density  of  a  saturated  vapor 
in  no  wise  depends  upon  the  size  or  shape  of  the  containing 
vessel;  and  this  is  known  to  be  the  fact.  It  is  also  inde- 
pendent of  the  area  of  the  free  surface  of  the  liquid,  though 
this,  perhaps,  may  not  be  so  evident.  If  the  free  surface  be 
doubled  (for  example),  we  shall  thereby  double  the  number 
of  both  the  out-going  and  the  in-coming  molecules  ;  hence  the 
equality  between  the  two  will  not  be  disturbed,  and  this  shows 
that  the  vapor  density  does  not  depend  upon  the  extent  of  the 
free  surface  of  the  liquid. 

Vapor  Pressure.  —  The  pressure  exerted  by  a  vapor  depends 
(1)  on  the  average  speed  of  the  molecules  composing  the  vapor, 
and  (2)  on  the  number  of  these  molecules  that  strike  against 
a  unit  area  of  the  containing  vessel,  per  second.  The  number 
of  molecules  that  strike  against  a  given  area  in  a  given  time 
will  depend  on  the  number  of  molecules  in  a  unit  volume  of 
the  vapor  —  that  is,  on  the  density  of  the  vapor  —  and  on  the 
average  molecular  velocity.  Hence  (since  the  average  molec- 


80       THE  MOLECULAR  THEORY  OF  MATTER. 

ular  velocity  is  a  function  of  the  temperature)  we  may  say 
that  the  pressure  exerted  by  a  vapor  on  the  walls  of  the  con- 
taining vessel  will  depend  on  (1)  the  density  of  the  vapor, 
and  (2)  its  temperature.  This  much  is  true  of  all  gaseous 
bodies,  as  I  have  already  explained  to  you  while  speaking  of 
gases  ;  but  in  the  case  of  a  saturated  vapor  in  contact  with 
its  liquid,  we  have  just  seen  that  the  density  of  the  vapor 
is  itself  a  function  of  the  temperature.  Hence  we  conclude 
that  the  pressure  exerted  by  such  a  vapor  against  the  vessel 
containing  it  depends,  ultimately,  only  upon  the  temperature 
of  the  vapor  and  its  liquid.  This  fact  has  long  been  known 
from  experiment,  and  many  attempts  have  been  made  to  find 
an  equation  which  should  represent  the  relation  between  the 
pressure  and  temperature  of  saturated  vapors.  Of  the  many 
equations  that  have  been  proposed,  B/ankine's  is  probably  as 
good  as  any.  He  found  that  the  relation  in  question  could 
be  represented  with  remarkable  accuracy  by  an  equation  of 
the  form 


a-- 

where  p  is  the  pressure,  t  the  absolute  temperature,  and  a, 
(3,  and  y  are  constant  quantities,  to  be  determined  for  each 
liquid  by  experiment.1* 

Ebullition.  —  Although  we  saw  that  the  density  of  a  satu- 
rated vapor  is  not,  strictly  speaking,  a  function  of  the  area  of 
the  free  surface  of  the  liquid,  it  must  nevertheless  be  borne 
in  mind  that  the  process  of  evaporation  is  of  such  a  nature 
that  it  cannot  take  place  unless  there  is  some  free  surface. 
For  it  consists  in  the  escape  of  certain  molecules  froin  a  free 
surface  ;  and  where  there  is  no  such  surface,  obviously  there 
can  be  no  evaporation.  It  is  not  essential  that  the  free  sur- 
face should  be  at  the  top  of  the  liquid.  For  example,  when 

*  For  the  values  of  a,  /S,  and  y  for  various  liquids,  see  Rankine's  Mis- 
cellaneous Scientific  Papers  (London,  Charles  Griffin  &  Company,  1881). 


EBULLITION.  81 

water  is  heated,  the  air  that  it  holds  in  solution  is  deposited 
in  small  bubbles  on  the  walls  of  the  containing  vessel ;  and 
evaporation  may  take  place  across  the  surface  of  these  bubbles, 
the  bubbles  increasing  in  size  as  they  fill  with  steam,  until 
presently  they  rise  to  the  surface  and  break.  If  water  be 
freed  from  such  dissolved  air,  by  protracted  boiling  or  other- 
wise, it  may  be  made  to  behave  in  a  remarkable  manner. 
Thus  Dufour  found  that  if  drops  of  water  so  prepared  are 
submerged  in  a  mixture  of  oil  of  cloves  and  linseed  oil  (of 
specific  gravity  1.000),  they  can  be  heated  far  above  the  boil- 
ing point,  although  exposed  only  to  atmospheric  pressure. 
Such  drops  have  no  free  surface,  and  therefore  no  evapora- 
tion can  take  place  from  them.  Dufour  heated  large  drops 
of  water,  in  this  way,  up  to  248°  Fahr.,  and  he  succeeded  in 
heating  smaller  ones  as  high  as  352°  Fahr.  (Under  ordinary 
circumstances  water  could  not  be  heated  to  352°  unless  it 
were  subjected  to  a  pressure  of  at  least  139  pounds,  absolute, 
to  the  square  inch.)  Wheft  these  drops  came  in  contact  with 
the  thermometer,  or  with  the  containing  vessel,  they  passed 
instantly  into  steam,  with  a  hissing  sound.  It  seems  probable 
that  the  explanation  of  such  phenomena  as  these  is,  that  when 
the  liquid  is  heated  gradually  and  uniformly,  without  any 
free  surface,  its  molecules  are  accelerated  throughout  the 
mass,  but  in  such  a  uniform  manner  that  the  momentum  is 
nowhere  sufficient  to  tear  them  apart  against  the  attractive 
forces  that  exist  among  them.  When  the  velocities  become 
so  great  that  in  parts  of  the  drop  they  are  on  the  point  of 
tearing  the  molecules  apart,  the  least  disturbance  from  with- 
out, such  as  a  shock  or  a  vibration  or  the  contact  of  some 
foreign  substance,  may  precipitate  the  disruption ;  and  after 
a  free  surface  has  once  been  formed,  even  though  it  may  be 
exceedingly  small,  the  drop  will  be  almost  instantly  dissipated 
by  evaporation  across  this  surface.  Ebullition  differs  from 
simple  evaporation  in  the  formation  of  such  free  surfaces  in 
the  interior  of  the  liquid,  or  along  the  bottom  and  sides  of 
the  containing  vessel,  across  which  surfaces  evaporation  occurs 


82       THE  MOLECULAR  THEORY  OF  MATTER. 

precisely  as  we  have  described  it  in  connection  with  the  upper, 
horizontal  surface,  or  in  connection  with  the  air-bubbles  sepa- 
rated from  solution  by  heat.  Ebullition  appears  to  occur  only 
when  heat  is  supplied  to  the  liquid  faster  than  it  can  diffuse 
to  the  surface  layers,  or  be  carried  there  by  convection.  The 
cause  of  the  formation  of  free  surfaces  in  the  interior  of  a 
boiling  liquid  is  not  well  understood  yet,  and  in  fact  it  may 
be  said,  in  general,  that  we  have  still  much  to  learn  concern- 
ing the  phenomena  of  ebullition,  both  by  the  making  of  new 
experiments,  and  by  interpreting  those  that  have  been  made 
already.* 

Critical  Points.  — Before  leaving  the  subject  of  vapors  and 
evaporation,  I  want  to  call  your  attention  to  a  peculiar  fact 
which  was  discovered  experimentally,  I  believe,  before  it  was 
explained  theoretically  ;  though  its  theoretical  explanation  is 
quite  simple.  I  refer  to  the  fact  that  there  is  a  temperature 
for  every  gas  —  called  its  critical  temperature  —  such  that 
the  gas  cannot  be  liquefied  by  pressure  alone  when  its  tem- 
perature exceeds  this  critical  value,  no  matter  how  great  the 
applied  pressure  may  be.  In  speaking  of  the  theory  of  evap- 
oration, I  called  your  attention  to  the  fact  that  a  molecule 
cannot  leave  its  liquid  unless  its  upward  velocity  exceeds  a 
certain  limit.  ISTow  if  molecules  attract  one  another,  there 
must  be  a  similar  proposition  true  of  molecules  that  collide 
with  one  another  in  a  vapor  or  a  gas.  Let  me  illustrate.  A 
stone  thrown  upward  by  the  hand  does  not  travel  far  before 
the  attractive  force  of  the  earth  upon  it  overcomes  its  momen- 
tum, and  causes  it  to  fall  back  again.  By  using  a  rifle  we 
can  project  a  ball  far  higher  into  the  air,  but  still  it  is  only 
a  matter  of  time  when  the  momentum  will  be  overcome,  and 
the  ball  will  fall  back  again  just  as  the  stone  did.  With  a 
good  modern  cannon  we  can  throw  a  projectile  several  miles 

*  In  connection  with  this  point,  the  reader  is  recommended  to  consult 
the  section  on  "Evaporation  and  Ebullition  "  in  Preston's  Theory  of  Heat 
(Macmillan  &  Co.,  1894). 


CRITICAL   POINTS.  83 

into  the  air,  —  and  still  it  falls  back.  But  it  is  conceivable 
that  we  might  project  one  with  such  speed  that  it  would  leave 
the  earth  forever.  Such  a  result  could  be  realized  without 
giving  the  projectile  an  infinite  velocity.  I  have  no  doubt 
that  Professor  Alden  will  show  you,  or  has  shown  you,  in 
your  course  in  mechanics,  that  if  the  retarding  action  of  the 
air  be  omitted  from  consideration,  an  initial  speed  of  36,650 
feet  per  second  would  be  quite  sufficient.  Now,  with  this 
much  premised,  let  us  imagine  two  molecules  of  a  gas  in 
contact ;  and  let  us  suppose  that  a  sudden  impulse  is  given 
to  one  of  them,  to  drive  it  away  from  the  other.  If  the 
impulse  is  small  enough,  the  disturbed  molecule  will  only 
travel  a  short  distance,  and  will  then  fall  back  to  its  original 
position;  but  we  may  give  it  such  a  speed  that  the  attractive 
force  of  the  fixed  molecule  will  fail  to  bring  it  back,  and  in 
this  case  it  will  travel  onward  indefinitely.  NOAV,  just  as  in 
the  case  of  the  cannon-ball  and  the  earth,  there  must  be  some 
intermediate  velocity  that  will  be  just  sufficient  to  separate  the 
two  molecules  under  consideration.  We  may  call  this  the 
critical  velocity ;  and  we  may  say,  that  if  the  molecules  of  a 
gas  are  moving  about  so  that,  on  an  average,  when  two  of 
them  collide  they  have  a  relative  velocity  greater  than  this 
critical  value,  the  gas  in  question  cannot  be  liquefied  by 
pressure  alone  j  for  even  if  its  molecules  were  forced  almost 
into  absolute  contact  with  one  another,  their  velocities  would 
be  sufficient  to  separate  them  again  indefinitely,  as  soon  as 
the  pressure  was  removed.  From  this,  and  from  the  rela- 
tion between  temperature  and  molecular  velocity  in  gases,  it 
follows  that  for  every  gas  there  is  a  temperature  above  which 
the  gas  cannot  be  liquefied.  The  critical  temperatures  of  the 
so-called  permanent  gases  are  very  low ;  and  that  is  why  they 
resisted  liquefaction  until  the  condition  necessary  to  success 
in  the  experiment  became  known.  The  following  table  gives 
the  critical  temperatures  of  certain  of  the  more  familiar  gases 
and  liquids : 


84 


THE   MOLECULAR   THEORY   OF   MATTER. 


TABLE  OF  CRITICAL  TEMPERATURES. 


CHEMICAL 

CRITICAL  TE 

MPERATURE. 

FORMULA. 

Centigrade. 

Fahrenheit. 

Hydrogen*      .     .  •  -.     .     . 
Nitrogen      . 
Oxv^en 

H 

N 

o 

-  229° 
-  124 
-  118 

-  380° 
-  191 
—  180 

Marsh  gas 

CH4 

—  100  (?) 

—  148  I?} 

Carbonic  acid  ..... 
Nitrous  oxide  ..... 
Ammonia  gas  .     .     ... 
Chlorine      ....... 
Sulphur  dioxide    .... 
Ether      *     .     . 

C02 

N2O 
NH3 
Cl 
S02 
(C2H5)20 

+    31 
+    36 
+  131 
+  141 
+  156 
+  194 

+    88 
+    97 
+  268 
+  286 
+  313 
+  381 

Alcohol  
Chloroform      
Carbon  disulphide     .     .     . 
Benzene      

C2H60 
CHC13 
CS2 
C6H6 

+  235 

-1-  260 

+  272 
+  281 

+  455 

-t-  500 
+  522 
+  538 

Acetic  acid 

C2H4O2 

+  322 

-1-  612 

Water                   . 

H90 

+  365 

+  689 

Contraction  and  Compressibility.  —  The  well-known  resist- 
ance of  liquids  to  compression  might  seem  to  indicate  that 
the  molecules  of  these  bodies  are  nearly  in  contact  with  one 
another;  and  the  assumption  that  they  are  in  contact  has 
sometimes  been  made,  for  the  purpose  of  deducing  their 
diameters  in  accordance  with  a  theorem  of  Clausius,  to 
which  I  shall  presently  refer.  The  fact  that  liquids  con- 
tract when  cooled  indicates,  on  the  other  hand.,  that  their 
molecules  are  not  in  contact.  I  think  I  can  tell  you  -how 
these  two  things  are  to  be  reconciled.  I  think  the  difficulty 
of  compressing  liquids  arises  from  the  fact  that  their  mole- 
cules are  describing  curved  paths  t  with  considerable  veloci- 
ties. They  would  fly  off  tangentially  to  these  paths  if  it 


*  Calculated  by  Mr.  Sutherland. 


t  See  Fig.  10. 


CONTRACTION   AND    COMPRESSIBILITY.  85 

were  not  for  the  inter-inolecular  attractive  forces  that  deter- 
mine the  curvature.  Normally,  therefore,  there  is  a  sort  of 
balance  between  the  attractive  forces  and  the  centrifugal 
tendency  due  to  the  velocity  of  the  molecules  and  to  the 
curvature  of  their  paths.  The  centrifugal  tendency  developed 
by  liquid  molecules  may  be  quite  considerable,  because  it  is 
proportional  to  the  square  of  the  velocity  of  translation,  and 
to  the  reciprocal  of  the  radius  of  curvature  of  the  path.  The 
radius  of  curvature  of  the  path  must  be  exceedingly  small, 
because  it  is  probably  of  the  same  order  of  magnitude  as  the 
distances  between  the  molecules.  Its  reciprocal,  therefore, 
will  be  correspondingly  large,  and  hence  the  liquid  molecules 
may  have  a  .considerable  centrifugal  tendency,  even  though 
their  velocity  of  translation  may  be  far  less  than  it  is  in 
gases.  Under  normal  conditions,  as  I  have  said,  the  attrac- 
tive forces  and  the  inertia  effects  are  balanced,  and  the  liquid 
remains  at  a  constant  volume.  When  we  cool  the  liquid  we 
diminish  its  kinetic  energy  —  that  is,  we  make  its  molecules 
go  slower.  This  lessens  the  centrifugal  tendency,  the  attrac- 
tive forces  preponderate,  and  the  molecules  approach  one 
another;  that  is,  the  liquid  contracts.  When,  instead  of 
cooling  the  liquid,  we  compress  it,  the  phenomena  are  not 
so  simple.  In  reducing  the  bulk  by  compression  we  cause  the 
molecules  to  approach  one  another.  This  lessens  their  poten- 
tial energy,  and  therefore  increases  their  kinetic  energy;  hence 
compression  increases  the  average  molecular  speed.  If  all 
the  other  circumstances  of  the  molecular  motion  remained 
the  same  as  before,  this  increase  of  speed  would  increase  the 
centrifugal  tendencies  of  the  molecules,  and  cause  the  liquid 
to  resist  compression.  But  in  compressing  a  liquid  we  do 
not  simply  accelerate  its  molecules.  In  bringing  them  nearer 
one  another  we  increase  the  attractive  forces  between  them, 
and  we  undoubtedly  alter  the  radii  of  curvature  of  their 
paths ;  and  to  predict  the  actual  behavior  of  the  liquid  we 
should  have  to  take  all  these  things  into  consideration.  The 
mathematical  treatment  of  this  problem  is  quite  difficult,  and 


86       THE  MOLECULAR  THEORY  OF  MATTER. 

I  cannot  positively  say  what  result  it  would  yield.  Never- 
theless I  am  satisfied,  in  my  own  mind,  that  in  forcing  the 
molecules  of  a  liquid  closer  together  we  increase  the  inertia 
effects  far  more  than  we  increase  the  attractive  forces,  and 
that  that  is  why  liquids  resist  compression  so  powerfully. 

Surface  Tension.  —  The  existence  of  molecular  forces  in 
liquids  may  be  readily  shown  by  experiment,  and  some  of  the 
experiments  that   have   been  devised   for   this  purpose  are 
exceedingly  beautiful  and  suggestive ;  but  before  describing 
any  of  them  I  wish  to  call  your  attention  to  a  fact  which 
enables  us   to  discuss  molecular  forces 
more   conveniently   than   we    otherwise 
could.     Consider,  for  a  moment,  a  drop 
of  liquid,  of  which  this  (Fig.  28)  is  an 
imaginary  sectional  view.     Particles  in 
the  interior  of  the  liquid  are  attracted 
equally  in  every  direction ;  but  particles 
on  the  surface  are  attracted  only  inward, 
in  a  direction  perpendicular  to  the  sur- 
face of  the  drop.    If  we  write  the  mathe- 
matical equations  that  would  represent 
FIG.  28. —  DIAGRAM  IL- 

LUSTRATING  SURFACE  the  behavior  of  a  drop  when  acted  upon 

FORCES  IN  A  DROP  OF  by  forces  of  this  kind,  we  shall  find  that 
LIQUID.  .  . 

these  equations  are  precisely  the  same 

as  they  would  be  in  a  similar  drop  whose  parts  do  not 
attract  one  another,  but  which  is  enveloped  by  a  water- 
tight skin,  or  membrane,  in  a  state  of  uniform  tension.  The 
tension  that  such  an  imaginary  membrane  would  have  to  have, 
to  produce  the  observed  phenomena  of  drops  and  other  small 
portions  of  liquid,  is  called  the  surface  tension  of  the  liquid. 
It  should  be  carefully  noted  that  we  do  not  assert  that  any 
such  membrane  actually  exists.  "  Surface  tension "  is  only  a 
convenient  conception  that  enables  us  to  foresee,  more  clearly, 
the  effects  produced  under  given  conditions  by  the  molecular 
forces  existing  in  liquid  masses.  I  think  I  need  not  discuss 


PHENOMENA   OF   FILMS.  87 

this  conception  further,  because  it  is  explained  and  illustrated 
in  all  good  works  on  physics.  It  may  be  well,  however,  to 
call  attention  to  the  fact  that  the  ideal  surface  membrane  of 
liquids  differs  from  all  material  membranes  inasmuch  as  its 
tension  does  not  increase  when  the  surface  of  the  liquid  is 
extended  in  any  way.  When  the  surface  of  a  liquid  increases, 
it  does  so  by  the  exposure  of  particles  that  were  previously  in 
the  interior,  and  not  by  the  stretching  of  the  old  surface,  in 
the  sense  in  which  the  word  " stretch'7  is  ordinarily  used. 
This  point  seems  obvious  enough,  but  I  find  that  unless 
particular  attention  is  called  to  it,  students  are  apt  to  get 
erroneous  ideas  about  surface  tension. 

Phenomena  of  Films. — Kegarding  liquids  as  mobile  bodies 
enclosed  in  contractile  membranes  in  a  state  of  uniform  ten- 
sion, it  will  be  plain,  I  think,  that  the  most  notable  effects  of 
molecular  attraction  will  be  observed  when  the  mass  of  the 
liquid  is  small  in  proportion  to  the  surface  it  exposes.  Thus 
the  phenomena  of  soap-bubbles,  and  of  other  forms  of  liquid 
films,  are  very  striking.  A  simple  and  beautiful  experiment 
illustrating  surface  tension  can  be  performed  with  a  piece  of 
wire,  a  bit  of  sewing  silk,  and  a  solution  of  soap,  suitable  for 
blowing  bubbles.3*  The  wire  is  bent  into  a  ring,  about  2-J- 
inches  in  diameter,  and  a  soap  film  is  formed  across  it  by 
simply  immersing  it  in  the  soap  solution  and  withdrawing 
it  again.  To  make  the  film  adhere,  it  may  be  necessary  to 
roughen  the  ring  with  a  file.  A  piece  of  sewing  silk,  about 
three  inches  long,  is  tied  together  at  the  ends,  so  as  to  form 
a  closed  loop.  The  silk  loop  is  then  wetted  with  the  soap 
solution,  and  laid  gently  on  the  film.  (See  Fig.  29.)  It  will 
lie  indifferently  in  any  position  if  the  film  be  kept  perfectly 
horizontal,  its  indifference  being  due  to  the  fact  that  the 
forces  acting  on  it  are  balanced  in  all  directions.  If  the 
film  be  broken  inside  the  silk  loop,  by  the  contact  of  a  dry 

*  Plateau's  liquide  glycerique  is  the  best  thing  for  blowing  bubbles. 
See  Appendix. 


88 


THE  MOLECULAR  THEORY  OF  MATTER. 


bit  of  wood  or  a  hot  needle,  the  loop  instantly  flies  out  into 
a  perfect  circle  (Fig.  30),  demonstrating  the  existence  of  the 
"surface  tension"  in  a  most  interesting  manner.  (The  same 


FIGS.  29  and  30. —ILLUSTRATING  THE  SILK-LOOP  EXPERIMENT. 

experiment  may  be  performed  with  the  ring  and  film  vertical, 
if  the  silk  loop  be  supported  as  shown  in  Fig.  31.)  When 
the  circular  silk  ring  is  floating  on  the  soap  film,  it  is  instruc- 
tive to  tap  it  lightly  on  the  inside  with  a  lead-pencil  or 

other  convenient  small  article. 
It  springs  away  from  the  touch 
of  the  pencil  as  though  it  were 
a  ring  of  tempered  steel  escap- 
ing from  the  blow  of  a  hammer. 
This  is  because  the  tap  with 
the  pencil  deforms  it  slightly, 
and  the  tension  of  the  film  im- 
mediately restoring  it  to  the 
circular  form  causes  it  to  react 
against  the  pencil  very  smartly, 
and  to  bound  away  from  it  with 
surprising  quickness.  An  end- 
less variety  of  beautiful  experiments  can  be  tried  with  soap 
films  with  extremely  simple  apparatus;  but  we  cannot  dwell 
upon  them  longer  this  evening.  The  one  experiment  that 


FIG.  31.  — MODIFICATION  OF  THE 
SILK-LOOP  EXPERIMENT. 


OTHER  SURFACE  PHENOMENA. 


89 


I  have  described  will  be  sufficient  to  show  the  existence  of  the 
molecular  forces  satisfactorily. 

•  Other  Surface  Phenomena.  —  The  surface  tension  of  a 
liquid  may  be  altered  by  a  variety  of  methods.  Thus  if  a 
small  sliver  of  cork  be  carefully  wetted  with  alcohol  along 
half  of  its  length  on  one  side,  and  be  then  thrown  upon  water, 
it  will  revolve,  because  the  surface  tension  of  dilute  alcohol 
is  less  than  the  surface  tension  of  pure  water,  and  the  cork  is 


FIG.  32.  — A  METHOD  FOB  OBTAINING  A  CLEAN  WATER-SURFACE. 

therefore  pulled  more  in  one  direction  than  the  other.  If  a 
similar  fragment  of  cork  be  wetted  with  alcohol  or  ether 
along  one  entire  end  or  side,  and  thrown  upon  water,  it  will 
be  pulled  across  the  surface  of  the  water  bodily,  for  the  same 
reason.  Particles  floating  on  the  surface  of  clean  water 
appear  to  be  repelled  by  a  drop  of  alcohol  or  ether,  brought 
near  to  them  on  the  end  of  a  glass  rod  or  the  tip  of  a  finger. 
This  is  because  some  of  the  vapor  of  the  alcohol  or  ether  con- 
denses on  the  surface  of  the  water  and  alters  its  tension  in 
the  vicinity  of  the  drop.  Small  particles  of  camphor  floating 


90       THE  MOLECULAR  THEORY  OF  MATTER. 

on  clean  water  exhibit  vigorous  movements.  Camphor  is 
slightly  soluble  in  water,  and  its  solution  has  a  lower  surface 
tension  than  pure  water.  The  camphor  particles  do  not  dis- 
solve with  equal  rapidity  on  all  sides,  and  hence  there  is  a 
resultant  pull  exerted  on  them,  in  the  direction  in  which  the 
water  contains  the  least  camphor  in  solution.  For  success  in 
this  experiment  it  is  essential  to  have  a  perfectly  clean  water 
surface,  as  the  least  trace  of  oily  matter  deadens  the  move- 
ments remarkably.  A  good  way  to  secure  such  a  surface  is 
shown  in  Fig.  32.  An  inverted  glass  funnel  is  connected 
with  the  faucet  by  a  rubber  tube,  and  the  water  is  allowed  to 
run  freely  for  some  time.  The  faucet  is  then  turned  off  until 
only  the  least  possible  amount  of  water  comes  through  it.  A 
piece  of  camphor  is  next  scraped  clean,  and  a  few  of  the  last 
scrapings  are  allowed  to  fall  on  the  water  in  the  funnel.  If 
the  funnel  was  originally  clean,  they  will  spin  about  in  a 
lively  manner.  We  shall  have  occasion  to  refer  to  this 
experiment  again,  when  we  come  to  consider  the  size  of 
molecules. 

Magnitude  of  the  Surface  Tension.  —  If  we  could  find  out 
what  the  strain  is  on  the  silk  thread  shown  in  Fig.  30,  we 
could  calculate  the  intensity  of  the  surface  tension  by  the 
same  formula  that  is  used  for  calculating  the  bursting  pressure 
of  thin,  hollow  cylinders.  It  would  be  possible  to  devise  an 
experiment  that  would  give  this  strain,  but  it  would  hardly 
be  worth  while,  because  the  surface  tension  of  liquids  can  be 
determined  by  other  methods  more  conveniently  and  with  far 
greater  accuracy.  In  fact,  it  can  be  determined  directly,  by 
means  of  an  apparatus  like  that  shown  in  Fig.  33,  where  S  is 
a  soap  film,  and  W  is  a  light  wire,  which  can  move  up  and 
down  without  sensible  friction.  By  putting  small  weights  in 
the  pan  we  can  readily  discover  what  the  supporting  power 
of  the  film  is.  This  result  is  to  be  divided  by  the  length  of 
that  part  of  the  wire  W  which  is  wetted  by  the  film,  and  the 
quotient  is  the  supporting  power  of  the  film,  per  unit  length 


MAGNITUDE    OF    THE    SURFACE    TENSION. 


91 


of  W.  To  find  the  surface  tension  we  have  to  divide  the 
supporting  power  thus  found  by  2,  because  the  film  has  two 
surfaces.  Experiments  of  this  sort  are  interesting  enough, 
but  they  are  of  small  value, 
because  they  can  only  give  us 
the  surface  tension  of  soap 
solutions,  or  of  other  liquids 
from  which  bubbles  can  be 
blown.  Different  methods 
have  to  be  used,  therefore,  to 
determine  the  surface  tension 
of  pure  water  and  the  great 
majority  of  liquids.  Many 
such  methods  have  been  pro- 
posed. One  of  them  is  based 
on  the  investigation  of  the 
curved  surface  of  a  liquid, 
where  it  touches  the  vessel 
Containing  it.  You  are  all  FIG.  33.  — APPARATUS  FOB  MEASURING 

familiar  with  the  appearance 

of  the   water   curve,   and   I  think  you   would  be   ready  to 

admit  that  the  experimental  investigation  of  its  shape  would 


FIG.  34. —THE  WATER-CURVE;  FROM  HAGEN'S  MEASURES. 

be  very  difficult.     The  difficulties  have  been  overcome,  how- 
ever, by  numerous  observers ;  and  I  have  plotted  for  you, 


92       THE  MOLECULAR  THEORY  OF  MATTER. 

from  Hagen's  data,  the  form  assumed  by  a  water  surface  in 
the  vicinity  of  a  flat,  vertical  plate  of  polished  brass  (Fig.  34).* 
The  dotted  line  shows  the  level  of  the  water  at  an  infinite 
distance  from  the  plate,  it  being  an  asymptote  to  the  water- 
curve.  It  can  easily  be  shown  that  if  y  is  the  height,  above 
the  dotted  line,  of  any  point,  P,  on  the  water-curve,  and  a  is 
the  angle  that  the  tangent  at  P  makes  with  the  dotted  line, 
then 


'  =  2\^- sin|,  (35) 

W  £ 

where  S  is  the  pull  exerted  by  the  water  surface  on  a  line  one 
unit  long,  and  w  is  the  weight  of  a  unit  volume  of  water. 
This  equation  is  derived  from  the  fact  that  the  curvature  of 
the  surface  at  any  point  must  be  just  sufficient  to  enable  the 
superficial  tension  to  support  a  column  of  water  of  height  y. 
You  will  see  that  if  we  measure  a  and  y  at  any  point,  and 
substitute  their  values  in  the  equation  I  have  just  given,  we 
can  calculate  S  (iv  being  a  known  quantity).  The  principal 
objection  to  this  method  is  the  difficulty  of  determining  the 
coordinates  of  the  water-curve  accurately.  A  very  simple 
method  of  finding  the  surface  tension  consists  in  observing 
the  height,  h,  to  which  a  liquid  will  rise  in  a  capillary  tube 
of  radius  r.  The  surface  tension,  S,  is  then  given  by  the 
equation  f 

s^  WTh 

2  COS  a 

where  w  is  the  weight  of  a  unit  volume  of  the  liquid,  and  a  is 
the  angle  which  the  surface  of  the  liquid  makes  with  the  tube 
at  the  point  where  the  two  come  into  actual  contact.  With 
a  clean  glass  tube  the  value  of  a  for  water  is  0°,  and  for 
mercury  it  is  128°  52'.  The  capillary-tube  method  gives 

*  The  scale  of  Fig.  34  is  such,  that  every  dimension  in  the  cut  is  ten 
times  the  corresponding  dimension  in  nature. 

t  For  the  derivation  of  this  equation  see  Maxwell's  article  on  Capillary 
Action  in  the  Encyclopaedia  Britannica. 


MAGNITUDE   OF   THE    SURFACE   TENSION. 


93 


fairly  satisfactory  results,  but  the  angle  a  varies  considerably 
with  the  degree  of  cleanliness  of  the  surface  of  the  glass 
tube,  and  it  would  be  desirable  to  have  a  good  method  that 
would  be  free  from  any  such  objection.  I  believe  that  Lord 
Kelvin's  method  meets  every  requirement  in  this  respect,  and 
although  I  have  not  tried  it,  and  do  not  know  what  results 


FIG.  35.  — LORD  KELVIN'S  APPARATUS  FOR 
DETERMINING  SURFACE  TENSION. 


FIG.  36. —NOZZLE  OF  LORD  KEL- 
VIN'S APPARATUS. 


have  been  obtained  from  it  by  others,  I  am  going  to  tell  you, 
briefly,  what  his  plan  is.*  His  apparatus  consists  of  two 
vessels,  A  and  B  (Fig.  35),  connected  by  a  flexible  syphon  in 
such  a  manner  that  the  level  of  the  liquid  in  B  can  be  varied 
at  will  by  raising  or  lowering  A.  At  the  bottom  of  B  there 
is  a  small  nozzle  with  a  capillary  opening  through  it  (shown 
on  an  enlarged  scale  in  Fig.  36).  The  liquid  in  B  runs  down 

*  See  his  Popular  Lectures  and  Addresses,  Vol.  I,  p.  45. 


UNIVERSITY 


94 


THE  MOLECULAR  THEORY  OF  MATTER. 


through  the  nozzle,  and  if  the  height  h  is  not  too  great,  a  drop 
will  form  at  the  bottom  of  the  nozzle,  whose  radius  of  curva- 
ture will  be  just  sufficient  to  enable  the  surface  tension  at 
that  point  to  sustain  a  column  of  liquid  of  height  h.  This 
radius  of  curvature  is  next  determined  by  any  of  the  methods 
given  in  the  books  on  physics  that  are  applicable  to  the  case 
—  some  optical  method  being  preferable  —  and  the  surface 
tension  is  then  given  by  the  equation 


pr      whr 

~ 


(36) 


where  r  is  the  radius  of  curvature  of  the  drop,  and  w  is  the 
weight  of  a  unit  volume  of  the  liquid.  In  his  article  on 
Capillary  Action  in  the  Encyclopedia  Britannica,  Maxwell 
gives  the  following  values  of  the  surface  tensions  of  various 
liquids  at  20°  C.  (68°  Fahr.),  as  determined  by  Quincke,  the 


TABLE  OF  SURFACE  TENSIONS  AT  20°  C. 


SURFACE 

TENSION. 

LIQUID. 

Specific 

Dynes 

Grammes 

Grains 

Gravity. 

per 

Centimeter. 

per 
Centimeter. 

per 
Inch. 

Water           .          .     . 

1  0000 

81 

083 

324 

Mercury 

13  5432 

540 

551 

O1     K.Q 

Bisulphide  of  carbon  . 

1.2687 

32.1 

.033 

1.28 

Chloroform  .... 

1.4878 

30.6 

.031 

1.22 

Alcohol    

0  7906 

25  5 

026 

1  02 

Olive  oil  

0.9136 

36.9 

.038 

1.47 

Turpentine        •     .     . 

0.8867 

29.7 

.030 

1.19 

Petroleum     .... 

0.7977 

31.7 

.032 

1.27 

Hydrochloric  acid  . 

1.1 

70.1 

.071 

2.80 

Solution   of    hyposul-  ) 
phite  of  sodium       ) 

1.1248 

77.5 

.079 

3.10 

LATENT  HEAT  OF  VAPORIZATION.          95 

free  surface  of  the  liquid  being  in  contact  with  air  in 
every  case.  He  gives  the  tensions  as  expressed  in  dynes  per 
linear  centimeter,  and  I  have  reduced  them  also  to  grammes 
per  linear  centimeter  and  to  grains  per  linear  inch  ;  so  that 
the  last  column  gives  the  normal  pull,  in  grains,  exerted  by  a 
liquid  on  a  straight  line,  one  inch  long,  lying  in  its  surface. 
(Of  course  there  is  an  equal  and  opposite  pull  on  the  other  side 
of  the  line,  unless  it  happens  to  form  one  of  the  boundaries 
of  the  surface,  as  shown  at  W  in  Fig.  33.) 

The  value  of  the  surface  tension  of  water  given  in  this 
table  is  certainly  too  great.  Brunner  found  it  to  be  75.2 
dynes,  and  Wolf  found  76.5  and  77.3.  Rayleigh's  deter- 
mination, based  on  a  study  of  the  wave-length  of  ripples, 
gave  73.9  dynes  at  18°  C.  The  latest  trustworthy  deter- 
mination that  I  have  seen  is  that  given  by  Mr.  T.  Proctor 
Hall  in  the  Philosophical  Magazine  for  November,  1893.  He 
finds  that  at  t°  centigrade  the  surface  tension  of  water,  in 
dynes  per  linear  centimeter,  is  75.48  —  .140  t,  which  is 
equivalent  to  .07694  —  .000143  J  when  expressed  in  grammes 
per  linear  centimeter. 

Latent  Heat  of  Vaporization.  —  To  vaporize  a  given  mass 
of  liquid  we  have  to  tear  all  its  molecules  apart,  in  opposition 
to  the  attractive  forces  that  exist  among  them,  and  we  have 
also  to  increase  the  average  speed  of  the  molecules.  Each  of 
these  operations  necessitates  the  expenditure  of  energy,  and 
the  total  amount  of  energy  thus  required  is  surprisingly  large. 
For  example,  to  vaporize  one  pound  of  water  (about  a  pint) 
at  212°  Fahr.,  we  have  to  expend  no  less  than  966  British 
units  of  heat,  or  about  753,000  foot-pounds  of  energy.  This 
shows  that  although  the  range  of  sensible  molecular  attrac- 
tion, in  water  and  other  liquids,  is  quite  small,  the  attrac- 
tive forces,  where  they  exist,  must  be  very  great.  Following 
are  the  latent  heats  of  vaporization  of  a  few  familiar 
liquids. 


96 


THE  MOLECULAR  THEOEY  OF  MATTER. 


LATENT  HEATS  or  VAPORIZATION.* 


LIQUID 

LAI 

ENT  HEAT. 

Heat  Units. 

Centimeter-Kilograms. 

Water    

535.9 

22,883 

Alcohol       

202  4 

8  642 

Bisulphide  of  Carbon  .     .     . 
Mercury 

86.7 
62  0 

3,702 
2  647 

Investigation  of  the  Work  done  in  bringing  a  Molecule  to 
the  Surface.  —  A  molecule  in  the  interior  of  a  liquid  is  always 
under  the  influence  of  the  attractive  forces  of  its  neighbors, 
and  the  resultant  force  exerted  upon  it  at  any  instant  must 
be  found  by  compounding  the  attractions  of  all  the  molecules 
that  are  near  enough  to  have  any  sensible  effect  upon  it.  The 
ceaseless  change  of  position  that  goes  on  among  the  molecules 
of  a  liquid  causes  this  resultant  force  to  assume  all  imaginable 
directions  and  magnitudes  in  rapid  succession ;  and  it  follows 
that  the  molecule  under  consideration  is  in  a  sort  of  statistical 
equilibrium,  if  I  may  use  the  phrase.  This  is  the  kind  of 
equilibrium  that  is  contemplated  when  we  say  that  molecules 
in  the  interior  of  a  liquid  are  attracted  equally  in  every  direc- 
tion. With  this  much  premised  I  am  going  to  tell  you  of  a 
property  of  liquids  that  we  shall  use,  presently,  in  estimating 
the  sizes  of  molecules.  The  property  that  I  refer  to  is  this : 
Whatever  the  law  of  molecular  attraction  may  be,  the  work 
that  has  to  be  done  against  molecular  attractive  forces  in  bring- 
ing a  particle  of  a  liquid  from  the  interior  to  the  surface  is 
precisely  one  half  of  the  work  that  would  have  to  be  done  to 
transport  that  particle  from  the  interior  of  the  liquid  to  an 
infinite  distance.  In  proving  this  proposition  we  shall  con- 

*At  the  boiling  points  of  the  respective  liquids.  A  gramme  of  the 
liquid  is  considered  in  each  case,  and  the  unit  of  heat  is  the  heat  required 
to  raise  the  temperature  of  one  gramme  of  water  from  3°  C.  to  4°  C. 


BRINGING  A  MOLECULE  TO  THE  SURFACE.     97 

ceive  a  sphere  to  be  described  about  the  particle  as  a  center, 
the  radius  of  the  sphere  (which  we  will  denote  by  R)  being 
such  that  the  attractions  of  all  the  particles  that  lie  outside 
of  it  can  be  neglected.  So  long  as  the  sphere  is  wholly  sub- 
merged, the  particle  is  in  equilibrium ;  but  when  the  distance 
of  the  particle  from  the  surface  of  the  liquid  becomes  less 
than  Rj  a  portion  of  the  sphere  will  project  above  the  liquid, 
and  the  particle  will  no  longer  be  in  equilibrium,  but  will  be 
attracted  downward,  towards  the  interior.  For  example,  con- 
sider this  diagram  (Fig.  37),  the  upper  part  of  which  illus- 


FIG.  37. —DIAGRAM  TO  ILLUSTRATE  THE  WORK  DONE  IN  BRINGING  A 
PARTICLE  OF  LIQUID  TO  THE  SURFACE. 

trates  six  successive  stages  of  the  approach  of  the  particle  to 
the  surface.  In  the  first  stage,  shown  on  the  left,  the  sphere 
just  touches  the  surface.  In  the  next  stage  a  part  of  the 
sphere  projects  above  the  liquid,  and  therefore  there  is  a 
resultant  downward  attraction  on  the  particle.  To  find  the 
magnitude  of  this  resultant,  conceive  a  horizontal  plane  to  be 
passed  through  the  lower  part  of  the  sphere  so  as  to  cut  off  a 
segment  equal  to  that  which  projects  above  the  surface  of  the 
liquid.  (The  plane  in  question  is  indicated  in  the  diagram 
by  a  short,  straight  line.)  It  follows  from  the  symmetry  of 
the  figure  that  the  component  attractions  of  all  the  molecules 
that  lie  above  this  plane  are  perfectly  balanced  ;  and  hence 
the  resultant  attraction  of  the  liquid  on  the  particle  we  are 
considering  is  equal  to  the  attraction  exerted  upon  it  by  the 


98       THE  MOLECULAR  THEORY  OF  MATTER. 

segment  that  the  imaginary  plane  cuts  off  from  the  lower  part 
of  the  sphere.  You  will  see  that  the  same  reasoning  applies 
at  every  stage,  until  the  particle  finally  lies  in  the  surface,  as 
shown  on  the  right.  Now,  if  you  please,  turn  your  attention 
to  the  lower  half  of  the  diagram,  which  represents  six  corre- 
sponding stages  in  the  removal  of  the  particle  from  ths  sur- 
face of  the  liquid  to  an  infinite  distance.  As  we  pass  from 
right  to  left,  the  particle,  in  each  position,  is  attracted  down- 
ward by  the  segment  of  the  sphere  which  lies  within  the 
liquid.  By  comparing  the  upper  and  lower  parts  of  the 
diagram,  you  will  see  (1)  that  the  particle  is  attracted  down- 
ward, whether  it  is  above  the  surface  or  below  it ;  and  (2) 
that  the  magnitude  of  the  attraction  is  the  same,  when  the 
particle  is  at  a  given  distance  from  the  surface,  whether  the 
particle  is  inside  the  liquid  or  outside.  It  follows  from  this 
symmetry  of  the  attractions,  that  the  work  required  to  bring 
the  particle  to  the  surface  of  the  liquid  is  precisely  equal  to 
the  work  required  to  carry  it  from  the  surface  to  a  point  out- 
side where  the  attraction  of  the  liquid  is  no  longer  sensible ; 
and  that  is  the  same  thing  as  saying  that  the  work  required 
to  bring  the  particle  to  the  surface  is  one  half  of  the  work 
required  to  transport  it  from  the  interior  of  the  liquid  to  an 
infinite  distance. 

Numerical  Estimation  of  the  Work  done  in  bringing  a 
Molecule  to  the  Surface. — We  are  now  in  position  to  com- 
pute the  work  that  would  have  to  be  expended,  against  inter- 
molecular  forces,  in  bringing  to  the  surface  of  any  given 
liquid  a  small  portion  of  it,  of  given  weight,  originally 
situated  in  the  interior.  For,  let  us  consider  what  happens 
when  a  unit  weight  of  the  liquid,  originally  at  (say)  20°  C., 
is  heated  to  its  boiling  point  and  then  entirely  evaporated. 
In  the  first  place,  the  mass  of  liquid  contains  a  certain  quan- 
tity of  kinetic  energy,  because  its  molecules  are  certainly  in 
motion.  We  cannot  calculate  this  energy,  because  we  do  not 
know  enough  about  the  velocities  of  molecules  in  liquids. 


BRINGING   A   MOLECULE   TO   THE    SURFACE.  99 

For  the  present,  therefore,  we  shall  merely  denote  it  by  ki. 
We  next  proceed  to  raise  the  temperature  of  the  liquid  from 
20°  C.  to  the  boiling  point.  In  doing  so  we  accelerate  the 
molecules  of  the  liquid,  and  (since  the  liquid  expands  upon 
being  heated)  we  also  do  a  certain  amount  of  work  in  pulling 
the  molecules  apart,  against  the  attractive  forces  they  exert 
upon  one  another.  The  kinetic  energy  that  has  been  added, 
in  increasing  the  speed  of  the  molecules,  will  be  denoted  by 
&2  >  and  the  total  amount  of  heat-energy  that  we  had  to  add 
in  order  to  raise  the  temperature  from  20°  C.  to  the  boiling 
point  will  be  denoted  by  h.  The  next  step  is  to  evaporate 
the  liquid  ;  and  in  order  to  effect  the  evaporation  we  have  to 
supply  the  quantity  of  heat  known  as  the  "latent  heat  of 
vaporization,"  which  we  shall  represent  by  H.  The  unit 
weight  of  liquid  has  now  been  transformed  into  a  unit  weight 
of  vapor,  and  obviously  we  have  the  relation 

h  +  H=  kz  +  kz+  W+  w,  (37) 

where  &2  is  the  kinetic  energy  added  in  heating  the  liquid 
from  20°  C.  to  the  boiling  point,  ks  is  the  kinetic  energy 
added  in  transforming  the  boiling-hot  liquid  into  vapor,  W 
is  the  total  energy  expended  in  overcoming  molecular  at- 
tractions, and  w  is  the  so-called  "  external  latent  heat  "  — 
that  is,  the  work  done  by  the  vapor  in  expanding  from 
the  liquid-volume  to  the  vapor-volume  against  atmospheric 
pressure.  Of  these  quantities  we  know  H,  h,  and  w,  and 
although  we  do  not  know  k2  and  ks  separately,  we  can  com- 
pute their  sum  with  a  sufficient  degree  of  approximation. 
In  fact, 


where  K  is  the  total  kinetic  energy  of  the  vapor,  which  can  be 
computed  when  we  know  the  number  of  degrees  of  freedom 
of  the  molecules,  and  their  average  velocity  of  translation. 
We  may  observe  that  kl  is  the  kinetic  energy  of  the  mole- 
cules in  the  liquid  state  ;  and  since  the  kinetic  energy  of  a 


100  THE   MOLECULAR   THEORY   OF   MATTER. 

body  varies  as  the  square  of  the  body's  velocity,  and  we  have 
good  reason  to  believe  that  the  molecular  velocity  in  a  liquid 
is  far  less  than  it  is  in  the  vapor  of  that  liquid,  it  follows 
that  &!  is  extremely  small  in  comparison  with  K.  We  shall 
therefore  neglect  it,  and  consider  kz-\-  k3  to  be  equal  to  K. 
With  this  modification  (37)  becomes  equivalent  to 

W=H+h  —  K—  w.  (38) 

The  quantities  H  and  h  are  given  in  the  books  on  heat, 
from  which  source  data  can  also  be  had  for  computing  w.  To 
determine  K  we  shall  treat  the  vapor  as  a  perfect  gas  ;  for 
although  this  mode  of  procedure  is  not  strictly  correct,  it  will 
give  results  accurate  enough  for  our  purpose.  We  have,  as 
the  kinetic  energy  of  translation  in  a  given  mass  of  vapor, 

(39) 


where  M  is  the  mass  of  the  vapor  and  u*  is  the  mean-square 
velocity  of  its  molecules,  as  defined  in  equation  (5).     But  we 

also  have 

w*A  *  ft 

p  =  -g->  (40) 

where  A,  being  the  absolute  density  of  the  vapor,  is  equal  to 
its  mass  divided  by  its  volume  —  that  is, 

M 

A=y 

Substituting  this  value  of  A  in  (40),  we  find  that 
which,  in  (39),  gives 


Now  k,  being  the  total  kinetic  energy  of  translation  of  the 
vapor,  involves  three  degrees  of  freedom  ;  and  hence  K,  the 

'TL 

total  kinetic  energy,  is  found  by  multiplying  k  by  —  >  where  n 

o 

*  See  equation  (13). 


BRINGING   A   MOLECULE   TO   THE   SURFACE.         101 


is  the  number  of  degrees  of  freedom  in  the  vapor  under  con- 
sideration. When  n  is  not  certainly  known  (as  in  most  cases 
it  is  not),  we  shall  be  obliged  to  substitute  for  it  its  value  in 
terms  of  y,  as  given  by  equation  (22).  I  am  aware  that  grave 
objections  could  be  urged  against  such  a  substitution  in  the 
case  of  vapors,  but  it  is  the  best  we  can  do.  It  gives  us 


n]c_ 
~  3  - 


2k      _   pV 


(41) 


By  computing  K  in  accordance  with  this  equation  for  the 
four  liquids  whose  latent  heats  of  vaporization  are  given  on 
page  96,  we  obtain  the  values  given  in  the  fourth  column  of 
the  accompanying  table. 

COMPUTATION  OF  THE  WORK   DONE   AGAINST   MOLECULAR  ATTRACTION 

IN    BRINGING   A    GRAMME    OF   LlQUID   TO    THE    SURFACE,    AT   20°   C. 


LIQUID 

H 

h 

K 

w 

W 

\W 

Water 

22,880 

3,420 

5910 

1,750 

18640 

9320 

Alcohol  

8,640 

1,690 

6,180 

630 

3,520 

1,760 

Bisulphide  of  Carbon     . 

3,700 

290 

1,790 

350 

1,850 

925 

Mercury      

2,650 

470 

390 

260 

2,470 

1,235 

All  of  the  quantities  in  the  table  are  expressed  in  centi- 
meter-kilograms, and,  as  the  heading  indicates,  the  computa- 
tion is  performed  for  one  gramme  of  liquid  in  each  case.  The 
various  quantities  are  given  only  approximately,  and  the 
results  have  the  same  order  of  certainty  (or  uncertainty)  as 
the  values  of  y  that  I  have  used  in  equation  (41).  The  quan- 
tity W  is  obtained  by  substituting  H,  h,  K,  and  w  in  equation 
(38) ;  and  as  Wis  the  work  that  would  have  to  be  expended, 
against  molecular  attraction,  in  order  to  transport  a  gramme 
of  the  liquid  from  the  interior  of  the  liquid  to  an  infinite 
distance  (taking  it  away  a  molecule  at  a  time),  it  follows, 
from  the  preceding  section,  that  ^  W  is  the  work  that  we 
should  have  to  do  in  order  to  bring  to  the  surface  a  gramme 


102  THE   MOLECULAR   THEORY   OF   MATTER. 

of  liquid  that  was  previously  in  the  interior.  I  have  dwelt 
upon  this  computation  at  greater  length  than  its  intrinsic 
importance  would  warrant,  because,  as  I  have  already  told 
you,  we  shall  presently  make  use  of  it  in  determining  the 
sizes  of  molecules. 


IV.     THE   MOLECULAR   THEORY   OF   SOLIDS. 

Condition  of  the  Theory.  —  The  molecular  theory  of  solids 
is  still  in  its  veriest  infancy ;  in  fact  it  might  almost  be  said 
that  there  is  no  such  theory.  The  phenomena  exhibited  by 
solids  are  extremely  complicated,  and  this  fact  implies,  with- 
out doubt,  a  corresponding  complexity  in  molecular  structure. 
In  order  to  give  you  a  clear  idea  of  the  formidable  task  that 
lies  before  the  philosopher  who  would  penetrate  the  inner 
secrets  of  the  solid  condition  of  matter,  let  me  quote  a  few 
words  from  Maxwell.  "  The  stress  [in  a  solid  body]  at  any 
given  instant,"  he  says,  "depends  not  only  on  the  strain  at 
that  instant,  but  on  the  previous  history  of  the  body.  Thus 
the  stress  is  somewhat  greater  when  the  strain  is  increasing 
than  when  it  is  diminishing,  and  if  the  strain  is  continued  for 
a  long  time,  the  body,  when  left  to  itself,  does  not  at  once 
return  to  its  original  shape,  but  appears  to  have  taken  a  set, 
which,  however,  is  not  a  permanent  set,  for  the  body  slowly 
creeps  back  towards  its  original  shape  with  a  motion  which 
may  be  observed  to  go  on  for  hours  and  even  weeks  after  the 
body  is  left  to  itself.  .  .  .  The  phenomena  are  most  easily 
observed  by  twisting  a  fine  wire  suspended  from  a  fixed  sup- 
port, and  having  a  small  mirror  suspended  from  the  lower 
end,  the  position  of  which  can  be  observed  in  the  usual  way 
by  means  of  a  telescope  and  scale.  If  the  lower  end  of  the 
wire  is  turned  round  through  an  angle  not  too  great,  and  then 
left  to  itself,  the  mirror  makes  oscillations,  the  extent  of 
which  may  be  read  off  on  the  scale.  These  oscillations  decay 
much  more  rapidly  than  if  the  only  retarding  force  were  the 
resistance  of  the  air,  showing  that  the  force  of  torsion  in  the 


ARRANGEMENT   OF   MOLECULES   IN    SOLIDS.         103 

wire  must  be  greater  when  the  twist  is  increasing  than  when 
it  is  diminishing.  This  is  the  phenomenon  described  by  Sir 
W.  Thomson  under  the  name  of  the  viscosity  of  elastic  solids. 
But  we  may  also  determine  the  middle  point  of  these  oscilla- 
tions, or  the  point  of  temporary  equilibrium  when  the  oscilla- 
tions have  subsided,  and  trace  the  variations  of  its  position. 
If  we  begin  by  keeping  the  wire  twisted,  say  for  a  minute  or 
an  hour,  and  then  leave  it  to  itself,  we  find  that  the  point  of 
temporary  equilibrium  is  displaced  in  the  direction  of  twist- 
ing, and  that  this  displacement  is  greater  the  longer  the  wire 
has  been  kept  twisted.  But  this  displacement  of  the  point  of 
equilibrium  is  not  of  the  nature  of  a  permanent  set,  for  the 
wire,  if  left  to  itself,  creeps  back  towards  its  original  position, 
but  always  slower  and  slower.  This  slow  motion  has  been 
observed  by  the  writer  going  on  for  more  than  a  week,  and 
he  has  also  found  that  if  the  wire  were  set  in  vibration  the 
motion  of  the  point  of  equilibrium  was  more  rapid  than  when 
the  wire  was  not  in  vibration.  We  may  produce  a  very  com- 
plicated series  of  motions  of  the  lower  end  of  the  wire  by 
previously  subjecting  the  wire  to  a  series  of  twists.  For 
instance,  we  may  first  twist  it  in  the  positive  direction,  and 
keep  it  twisted  for  a  day,  then  in  the  negative  direction  for 
an  hour,  and  then  in  the  positive  direction  for  a  minute. 
When  the  wire  is  left  to  itself  the  displacement,  at  first  posi- 
tive, becomes  negative  in  a  few  seconds,  and  this  negative 
displacement  increases  for  some  time.  It  then  diminishes, 
and  the  displacement  becomes  positive,  and  lasts  a  longer 
time,  till  it,  too,  finally  dies  away."*  I  think  I  need  not  add 
anything  to  the  passage  I  have  just  quoted.  The  facts  to  be 
explained  by  the  molecular  theory  of  solids  are  certainly  for- 
bidding enough,  and  it  is  not  strange  that  so  little  progress 
has  been  made. 

Arrangement    of    the    Molecules  in  Solids.  —  The  most 
obvious  property  of  a  solid  is,  that  it  preserves  its  shape  so 

*  Encyclopaedia  Britannica,  article  Constitution  of  Bodies. 


104      THE  MOLECULAR  THEORY  OF  MATTER. 

long  as  it  is  not  acted  upon  by  external  forces.  Moreover, 
when  such  forces  are  applied,  the  solid  indeed  becomes 
deformed,  but  it  eventually  regains  its  original  shape  after 
the  forces  have  been  removed,  provided  they  did  not  exceed  a 
certain  magnitude  called  the  "  elastic  limit,"  which  is  peculiar 
to  the  solid  under  examination  and  to  the  way  in  which  the 
forces  were  applied.  We  are  obliged  to  conclude,  from  these 
facts,  that  the  molecules  of  a  solid  are  not  free  to  roam  about, 
but  that  some  or  all  of  them  have  determinate  mean  positions 
about  which  they  may  oscillate  and  rotate,  but  from  which 
they  never  permanently  depart  except  when  constrained  to 
do  so  by  an  external  force  greater  than  whatever  the  forces 
may  be  that  determine  the  mean  positions.  In  order  to  gain 
a  clearer  insight  into  the  nature  of  solids,  let  us  now  regard 
them  from  a  somewhat  different  point  of  view  —  let  us  con- 
sider the  transition  of  a  liquid  into  a  solid  by  congelation. 
It  is  well  known  that  when  a  liquid  is  cooled  to  a  certain 
point  peculiar  to  itself,  it  solidifies,  changing  its  volume  at 
the  same  time  and  giving  out  a  certain  amount  of  heat,  called 
the  "latent  heat  of  fusion."  We  are  therefore  impelled  to 
believe  that  when  the  average  kinetic  energy  of  translation 
of  the  liquid  molecules  is  reduced  to  a  certain  value  by  the 
abstraction  of  heat,  the  inter-molecular  attractive  forces 
become  able  to  restrict  the  migrations  of  the  molecules,  and 
to  confine  them,  as  I  have  already  said,  to  certain  mean 
positions.  Furthermore,  since  energy  cannot  be  created  or 
destroyed,  the  fact  that  the  body  gives  up  a  certain  amount 
of  "  latent  heat "  when  it  solidifies  must  necessarily  imply 
that  there  has  been  a  diminution  in  the  kinetic  energy  of  the 
molecules,  or  in  their  potential  energy,  or  in  both.  Now 
there  undoubtedly  is  some  loss  of  kinetic  energy  when  a  liquid 
solidifies,  but  I  think  we  can  assert  that  there  is  a  loss  of 
potential  energy  also,  just  as  there  is  when  a  gas  condenses 
into  a  liquid.  If  all  liquids  contracted  upon  solidifying  it 
would  be  plain  that  potential  energy  disappears  during  the 
process ;  for  contraction  would  imply  that  the  molecules 


ARRANGEMENT   OF    MOLECULES   IN   SOLIDS.         105 

approach  one  another.  The  fact  is,  however,  that  many 
liquids  expand  upon  solidification,  water  being  a  familiar 
example  of  this.  Hence  we  are  called  upon  to  explain  how 
it  can  be  that  the  potential  energy  in  a  body  can  grow  less 
when  the  molecules  of  the  body  go  farther  apart.  I  believe 
that  this  question  has  a  close  bearing  on  the  constitution  of 
molecules  —  a  subject  of  which  we  know  practically  nothing, 
at  present.  If  you  will  glance  at  these  diagrams,  however, 
(Figs.  38  and  39,)  I  think  you  will  see  that  the  condition  of 


FIGS.  38  and  39.  —ILLUSTRATING  MINIMUM  POTENTIAL  ENERGY  AND 
MINIMUM  VOLUME. 

least  potential  energy  in  a  system  of  bodies  does  not  neces- 
sarily coincide  with  that  of  closest  approach  of  the  centers  of 
those  bodies.  The  diagrams  are  not  intended  to  represent 
molecules  in  any  sense,  but  they  may  serve  to  illustrate  the 
point  under  consideration.  I  have  assumed  that  each  of  these 
ideal  bodies  has  four  attractive  poles,  which  are  represented 
by  the  black  spots.  The  centers  of  attraction,  or  poles,  of 
such  bodies  would  tend  to  approach  one  another  as  closely  as 
possible,  and  the  potential  energy  of  the  system  would  be 
least  when  the  poles  were  nearest  together  —  that  is,  when 
the  arrangement  was  like  that  shown  in  Fig.  38.  If  the 
bodies  were  placed  as  shown  in  Fig.  39  the  total  volume 
would  be  least,  but  the  potential  energy  in  that  case  would 
be  greater  than  in  the  arrangement  shown  in  Fig.  38,  because 


106      THE  MOLECULAR  THEORY  OF  MATTER. 

the  attractive  poles  are  further  apart.  It  is  plain,  therefore, 
that  the  potential  energy  of  a  system  of  bodies  is  not  neces- 
sarily least  when  the  space  occupied  by  the  system  is  least. 
In  other  words,  the  equipotential  surfaces  of  molecules  are 
not  necessarily  spherical.  This  conclusion,  derived  from  a 
consideration  of  the  phenomena  of  congelation,  is  confirmed 
by  a  study  of  the  physical  properties  of  crystals  with  unequal 
axes. 

Maxwell's  Views  Concerning  the  Molecular  Constitution  of 
Solids.  —  In  speaking  of  the  classification  of  bodies  under  the 
three  heads  of  solids,  liquids,  and  gases,  I  said,  in  the  early 
part  of  the  evening,  that  such  a  division  is  not  entirely  satis- 
factory ;  and  we  have  now  come  to  a  point  where  we  must 
examine  this  classification  more  carefully.  Many  bodies  are 
undoubtedly  solids,  and  many  others  are  undoubtedly  liquids  ; 
but  there  are  bodies,  such  as  wax,  pitch,  and  tar,  which 
possess  some  of  the  properties  of  each,  and  which  are  quite 
difficult  to  classify  satisfactorily.  A  mass  of  pitch,  for 
example,  may  be  brittle,  so  that  it  is  easily  shattered  by  a 
blow,  and  yet  this  same  mass,  when  placed  on  an  inclined 
plane,  loses  its  shape  and  flows  slowly  down  the  plane.  It 
may  not  reach  the  bottom  for  months,  or  perhaps  years ;  but 
a  true  solid  would  remain  where  it  was  placed,  for  all 
eternity.*  To  account  for  the  properties  of  pitch-like  bodies, 
and  for  the  peculiar  behavior  of  wires  and  other  solids  when 
submitted  to  strains,  Maxwell  has  proposed  the  following 
hypothesis:  "We  know  that  the  molecules  of  all  bodies  are 
in  motion.  In  gases  and  liquids  the  motion  is  such  that  there 
is  nothing  to  prevent  any  molecule  from  passing  from  any 
part  of  the  mass  to  any  other  part ;  but  in  solids  we  must 
suppose  that  some,  at  least,  of  the  molecules  merely  oscillate 
about  a  certain  mean  position,  so  that,  if  we  consider  a  certain 
group  of  molecules,  its  configuration  is  never  very  different 

*  It  is  assumed,  of  course,  that  the  inclination  of  the  plane  is  not  great 
enough  for  either  body  to  slide  upon  it. 


THE   MOLECULAR   CONSTITUTION    OF    SOLIDS.        107 

from  a  certain  stable  configuration,  about  which  it  oscillates. 
This  will  be  the  case  even  when  the  solid  is  in  a  state  of 
strain,  provided  the  amplitude  of  the  oscillations  does  not 
exceed  a  certain  limit,  but  if  it  exceeds  this  limit  the  group 
does  not  tend  to  return  to  its  former  configuration,  but  begins 
to  oscillate  about  a  new  configuration  of  stability,  the  strain 
in  which  is  either  zero,  or  at  least  less  than  in  the  original 
configuration.  The  condition  of  this  breaking  up  of  a  con- 
figuration must  depend  partly  on  the  amplitude  of  the  oscil- 
lations, and  partly  on  the  amount  of  strain  in  the  original 
configuration ;  and  we  may  suppose  that  different  groups  of 
molecules,  even  in  a  homogeneous  solid,  are  not  in  similar 
circumstances  in  this  respect.  Thus  we  may  suppose  that  in 
a  certain  number  of  groups  the  ordinary  agitation  of  the 
molecules  is  liable  to  accumulate  so  much  that  every  now  and 
then  the  configuration  of  one  of  the  groups  breaks  up,  and 
this  whether  it  is  in  a  state  of  strain  or  not.  We  may  in  this 
case  assume  that  in  every  second  a  certain  proportion  of  the 
groups  break  up,  and  assume  configurations  corresponding  to 
a  strain  uniform  in  all  directions.  If  all  the  groups  were  of 
this  kind,  the  medium  would  be  a  viscous  fluid.  But  we  may 
suppose  that  there  are  other  groups,  the  configuration  of 
which  is  so  stable  that  they  will  not  break  up  under  the 
ordinary  agitation  of  the  molecules  unless  the  average  strain 
exceeds  a  certain  limit,  and  this  limit  may  be  different  for 
different  systems  of  these  groups.  Now  if  such  groups  of 
greater  stability  are  disseminated  through  the  substance  in 
such  abundance  as  to  build  up  a  solid  framework,  the  substance 
will  be  a  solid,  which  will  not  be  permanently  deformed 
except  by  a  stress  greater  than  a  certain  given  stress.  But  if 
the  solid  also  contains  groups  of  smaller  stability  and  also 
groups  of  the  first  kind  which  break  up  of  themselves,  then 
when  a  strain  is  applied  the  resistance  to  it  will  gradually 
diminish  as  the  groups  of  the  first  kind  break  up,  and  this 
will  go  on  till  the  stress  is  reduced  to  that  due  to  the  more 
permanent  groups.  If  the  body  is  now  left  to  itself,  it  will 


108      THE  MOLECULAR  THEORY  OF  MATTER. 

not  at  once  return  to  its  original  form,  but  will  only  do  so 
when  the  groups  of  the  first  kind  have  broken  up  so  often  as 
to  get  back  to  their  original  state  of  strain.  This  view  of  the 
constitution  of  a  solid,  as  consisting  of  groups  of  molecules 
some  of  which  are  in  different  circumstances  from  others,  also 
helps  to  explain  the  state  of  the  solid  after  a  permanent 
deformation  has  been  given  to  it.  In  this  case  some  of  the 
less  stable  groups  have  broken  up  and  assumed  new  configura- 
tions, but  it  is  quite  possible  that  others,  more  stable,  may 
still  retain  their  original  configurations,  so  that  the  form  of 
the  body  is  determined  by  the  equilibrium  between  these  two 
sets  of  groups ;  but  if,  on  account  of  rise  of  temperature, 
increase  of  moisture,  violent  vibration,  or  any  other  cause, 
the  breaking  up  of  the  less  stable  groups  is  facilitated,  the 
more  stable  groups  may  again  assert  their  sway,  and  tend  to 
restore  the  body  to  the  shape  it  had  before  its  deformation.'77  * 
I  have  quoted  Maxwell's  words  at  length,  both  because  he  has 
stated  his  views  with  remarkable  lucidity,  and  because  his 
hypothesis,  so  far  as  I  know,  is  the  only  one  yet  proposed 
that  is  at  all  adequate  to  explain  the  complicated  phenomena 
exhibited  by  solid  bodies.  Even  here  we  should  proceed  with 
caution,  however,  for  the  hypothesis  that  he  offers  us,  pro- 
found and  beautiful  as  it  is,  cannot  logically  be  regarded  as 
anything  more  than  a  hypothesis  until  its  consequences  have 
been  worked  out  mathematically,  and  compared  with  the 
facts. 

•% 

Sublimation.  —  Some  solids  exhibit  a  phenomenon  quite 
analogous  to  evaporation,  to  which  phenomenon  the  name  of 
sublimation  has  been  given.  "  There  are  doubtless  frequent 
collisions  among  the  molecules  of  solids,  preciselyas  there 
are  among  those  of  liquids  and  gases.  "Such  being  the  case, 
we  must  conclude  that  the  velocities  of  oscillation  and  rotation 
of  the  molecules  of  solids  are  not  all  equal ;  and  doubtless 
there  is  also  some  undiscovered  law  of  distribution  of  these 

*  Encyclopaedia  Britannica,  article  Constitution  of  Bodies. 


SUBLIMATION.  109 

velocities,  analogous  to  Maxwell's  law  for  gases.  If  this  be 
admitted,  we  may  infer  that  every  now  and  then  a  molecule 
on  the  exterior  of  a  solid  will  have  an  impetus  sufficient  to 
overcome  the  restraining  forces,  and  to  tear  itself  away  and 
escape  from  the  solid,  just  as  molecules  escape  from  a  liquid 
in  evaporation.  "With  some  solids,  such  as  camphor,  this 
action  is  very  noticeable.  Ice  also  evaporates  (or  sublimes), 
in  dry  air,  at  temperatures  below  the  melting  point.  These 
two  substances,  as  well  as  others,  even  have  a  definite  vapor 
pressure  corresponding  to  each  temperature,  just  as  liquids  do 
when  placed  in  closed  vessels.  The  reason  for  this  definite 
relation  between  temperature  and  vapor  pressure  need  not  be 
repeated,  for  it  is  precisely  the  same  as  I  have  already  given 
you  in  speaking  of  evaporation  from  liquids.  '  Many  solids, 
such  as  iron  and  stone,  do  not  sensibly  sublime  at  any  tem- 
perature, though  metals  and  argillaceous  earths  and  other 
non-volatile  solids  often  have  characteristic  odors  which  may 
possibly  be  due  to  a  slight  loss  of  substance  by  sublimation. 
Other  solids,  such  as  arsenic,  sublime  freely  at  elevated  tem- 
peratures, mercury  bichloride,  or  "  corrosive  sublimate/7  taking 
its  popular  name  from  this  property.  There  are  reasons  for 
believing  that  even  carbon  is  slightly  volatile  at  high  tem- 
peratures—  at  least  under  the  conditions  to  which  it  is 
exposed  in  the  bulbs  of  incandescent  electric  lamps.  "*  It  must 
be  admitted  that  solids,  in  general,  do  not  sublime  as  freely 
as  one  might  naturally  expect  them  to ;  and  the  reason  for 
this  probably  is,  that  the  intermolecular  forces  in  these  bodies 
are  so  enormous  that  it  is  seldom  that  a  molecule  happens  to 
have  a  velocity  great  enough  to  enable  it  to  fly  away.  (That 
the  attractive  forces  between  neighboring  molecules  are  great, 
in  solids,  is  shown  by  the  tensile  strength  of  these  bodies.) 
When  the  intermolecular  forces  in  solids  are  counterbalanced 
in  some  degree  by  other  forces,  acting  in  opposition  to  them, 
there  is  often  a  loss  of  molecules  from  the  solid,  and  the 
resulting  phenomena  correspond,  to  some  extent,  with  those 
of  sublimation.  Examples  of  this  are  afforded  by  the  phenom- 


110      THE  MOLECULAR  THEORY  OF  MATTER. 

enon  known  as  solution,  and  also  by  that  known  as  dissociation. 
We  shall  consider  dissociation  first. 

Dissociation.  —  If  steam  be  passed  through  a  tube  contain- 
ing red-hot  iron-filings,  the  steam  is  decomposed,  black  oxide 
of  iron  is  formed,  and  hydrogen  escapes  from  the  free  end  of 
the  tube.  On  the  other  hand,  if  hydrogen  be  passed  through 
a  red-hot  tube  containing  black  oxide  of  iron,  the  oxide  is 
reduced  to  metallic  iron,  the  liberated  oxygen  combines  with 
the  hydrogen,  and  steam  escapes  from  the  free  end  of  the 
tube.  These  apparently  contradictory  phenomena  were  first 
adequately  explained  by  Deville  ;  and  in  order  to  understand 
his  explanation  properly,  let  us  first  consider  what  forces  are 
involved  in  the  problem.  (1)  The  hydrogen  molecule  being 
double,  we  have  to  consider  the  attraction  of  its  two  compo- 
nent atoms  for  each  other.  (2)  We  have  likewise  to  consider 
the  attraction  that  the  atoms  composing  an  iron  molecule 
exert  upon  one  another,  and  also  (3)  the  attractions  existing 
among  the  atoms  of  a  steam  molecule,  and  (4)  those  existing 
among  the  atoms  of  a  molecule  of  the  oxide  of  iron.  Let  us 
now  return  to  the  first  experiment,  in  which  steam  is  passed 
over  red-hot  iron.  In  this  case  the  motions  of  the  steam 
molecules  are  accelerated  by  the  heat  to  s-uch  a  degree  that 
some  of  them  are  torn  asunder,  and  oxygen  and  hydrogen 
atoms  are  liberated,  and  mingle  with  the  molecules  of  steam. 
If  no  iron  were  present,  these  dissociated  atoms  of  oxygen 
and  hydrogen  would  occasionally  chance  to  collide  with  one 
another  again,  and  in  some  of  these  cases  the  speed  of  the 
colliding  atoms  would  not  be  great  enough  to  prevent  them 
from  recombining  and  forming  new  molecules  of  steam.  The 
steam-molecules  are  therefore  always  dissociating  and  re-form- 
ing, and  it  is  plain  that  at  any  given  instant  the  proportion 
of  such  molecules  that  are  in  a  state  of  dissociation  will  be 
greater,  the  higher  the  average  speed  of  the  steam-molecules 
—  that  is,  the  higher  the  temperature.  As  a  matter  of  fact, 
however,  some  of  the  liberated  oxygen  atoms  come  in  contact 


DISSOCIATION.  Ill 

with  the  iron  that  is  present,  and  combine  with  it  to  form  the 
black  oxide  ;  and  the  hydrogen  atoms  that  were  formerly 
their  partners  pass  on  with  the  steam.  In  the  second  experi- 
ment the  phenomena  are  very  similar.  The  heat  partially 
dissociates  the  iron  oxide,  the  oxygen  atoms  thus-  liberated 
find  partners  among  the  hydrogen  atoms  flowing  overhead, 
and  the  steam  thus  formed  is  swept  away  in  the  stream  of 
hydrogen,  and  is  prevented  from  again  coming  in  contact  with 
the  reduced  iron  that  has  been  left  behind.  Now  let  us  con- 
sider the  experimental  tube  to  be  stopped  up  at  both  ends,  so 
that  nothing  can  enter  it  or  leave  it.  All  the  various  phenom- 
ena I  have  described  will  then  take  place  simultaneously,  and 
a  permanent  distribution  will  presently  be  reached,  in  which 
molecules  of  iron  oxide  are  dissociating  in  some  places  just 
as  fast  as  molecules  of  oxygen  are  combining  with  metallic 
iron  in  other  places,  and  molecules  of  steam  are  dissociating 
just  as  fast  as  other  molecules  of  steam  are  forming.  When 
this  condition  of  equilibrium  is  established,  the  composition 
of  the  contents  of  the  tube  will  appear  to  be  constant ;  but 
the  constancy  is  only  apparent  —  it  is  of  a  statistical  nature, 
just  as  the  constancy  of  the  vapor-density  over  a  liquid  is 
statistical.  If  the  distribution  of  the  various  substances  in 
the  experimental  tube  is  altered,  the  balance  will  be  destroyed. 
For  example,  if  some  of  the  hydrogen  is  removed,  more  steam 
will  be  dissociated  and  more  iron  oxide  formed,  until  the 
proportion  of  hydrogen  to  steam  becomes  the  same  as  before. 
Conversely,  if  some  of  the  steam  is  removed  the  iron  oxide 
will  be  dissociated  faster  than  it  is  formed,  the  oxygen  thus 
liberated  will  combine  with  some  of  the  hydrogen,  and  more 
steam  will  be  formed  ;  and  this  readjustment  will  continue 
until  the  original  proportion  of  hydrogen  to  steam  is  again 
restored.  Hence  it  is  plain  that  if  the  steam  that  is  formed 
is  continually  swept  away  by  a  current  of  hydrogen,  the  result 
will  be  that  the  oxide  of  iron  will  be  all  ultimately  reduced  ; 
while  if  the  hydrogen  that  is  liberated  is  swept  away  by  a 
current  of  steam,  the  iron  will  be  continuously  oxidized,  until 


112      THE  MOLECULAR  THEORY  OF  MATTER. 

it  is  all  converted  into  oxide.  This  beautiful  explanation  of 
the  phenomena  is  due  to  Deville,  and  the  principles  that 
underlie  his  theory  have  been  applied  to  a  wide  range  of  other 
phenomena,  and  upon  them  a  complete  theory  of  chemical 
equilibrium  has  been  erected.  I  regret  that  time  will  not 
allow  me  to  discuss  this  most  interesting  subject  further,  nor 
even  to  give  more  examples  of  dissociation  ;  but  you  will  find 
Guldberg  and  Waage's  generalized  theory  of  chemical  equi- 
librium given  in  Ostwald's  Outlines  of  General  Chemistry. 

Solutions.*  —  Many  solids,  when  brought  into  contact  with 
certain  liquids  under  suitable  conditions,  cease  to  exist  as 
solids,  and  diffuse  throughout  the  liquid  in  which  they  are 
submerged.  This  process,  by  which  the  solid  is  caused  to 
disappear,  is  called  solution;  and  the  same  word  is  also  used, 
as  a  noun,  to  signify  the  liquid  that  results  from  the  process. 
In  some  cases  the  liquid  acts  chemically  upon  the  solid,  form- 
ing a  new  compound,  which  remains  behind  when  the  solvent 
is  evaporated  —  an  instance  of  such  action  being  the  formation 
of  chloride  of  sodium  when  caustic  soda  is  placed  in  hydro- 
chloric acid.  We  shall  not  discuss  these  cases,  but  shall  con- 
fine our  attention  exclusively  to  those  in  which  the  solvent 
is  not  positively  known  to  act  chemically  upon  the  dissolved 
substance  —  as  when  sugar  is  dissolved  in  water,  or  silver 
chloride  in  dilute  ammonia.  In  such  instances  we  have  to 
think  of  the  molecules  of  the  liquid  as  exerting  a  certain 
attraction  on  the  external  molecules  of  the  solid ;  though  we 

*  In  accordance  with  the  general  plan  of  this  book,  I  have  included, 
in  the  present  section,  only  such  points  as  seem  to  be  well  established, 
and  to  have  an  immediate  bearing  on  the  molecular  structure  or  deport- 
ment of  bodies.  Our  knowledge  of  solutions  is  still  far  from  perfect, 
and  the  reader  will  find  that  chemists  are  divided,  upon  this  subject,  into 
two  great  schools  or  factions.  I  will  not  pretend  to  decide  between  these 
factions.  The  reader  may  profitably  consult  the  article  on  Solutions  in 
Watts's  Dictionary  of  Chemistry  (last  edition,  1894),  where  he  will  find 
the  rival  doctrines  ably  expounded.  Ostwald's  Solutions  (London,  Long- 
mans, Green  &  Co.,  1891)  can  also  be  recommended. 


SOLUTIONS.  113 

shall  be  careful  not  to  say  much  about  the  ultimate  nature  of 
this  attraction,  lest  we  might  give  offense  to  the  advocates  of 
one  or  other  of  the  two  schools  into  which,  on  this  point, 
chemists  are  divided.  If  the  attraction  of  the  liquid  for  the 
molecules  of  the  solid  is  slight,  the  attraction  of  the  molecules 
of  the  solid  for  one  another  may  still  greatly  preponderate, 
and  in  that  case  there  will  be  no  solution,  and  we  shall  have 
merely  a  solid  submerged  in  a  liquid.  On  the  other  hand,  if 
the  external  molecules  of  the  solid  are  attracted  outward  by 
the  liquid  more  powerfully  than  they  are  attracted  inward  by 
the  solid  itself,  they  will  be  detached,  and  the  solid  will 
rapidly  dissolve.  In  case  the  attraction  towards  the  liquid  is 
less  than  that  towards  the  solid,  but  yet  comparable  with  it, 
particles  will  tear  themselves  loose  from  the  solid  and  pass 
into  solution,  whenever  their  velocities  chance  to  be  great 
enough  to  enable  them  to  overcome  the  excess  of  attraction 
towards  the  solid.  In  some  cases  heat  will  be  evolved  during 
the  process  of  solution,  and  in  other  cases  it  will  disappear, 
or  become  latent ;  and  we  cannot  say,  in  advance,  which  of 
these  phenomena  will  take  place  in  any  given  instance,  until 
we  fully  understand  the  nature  and  magnitude  of  the  various 
forces  that  are  called  into  operation.  I  believe  that  many  of 
the  phenomena  of  solutions,  as  well  as  those  of  crystallization 
and  other  changes  of  state  in  bodies,  can  be  explained  by 
means  of  the  single  assumption  that  in  every  case  the  mole- 
cules of  the  system  tend  towards  that  configuration  in  which 
their  united  potential  energy  is  as  small  as  possible,  just  as  a 
marble  when  thrown  into  a  wash-bowl  tends  continually 
towards  the  bottom  of  the  bowl,  where  its  potential  energy  is 
least.  Thus,  when  we  find  that  a  given  solid  dissolves  in  a 
given  liquid,  we  are  to  infer  that  the  potential  energy  of  the 
system  is  less  when  the  solid  is  in  solution  than  it  is  when 
the  solid  is  intact,  and  merely  submerged  in  the  pure  liquid ; 
and  when  we  find  that  a  given  solid  does  not  dissolve  in  a 
given  liquid,  the  converse  is  to  be  inferred.  I  can  imagine 
that  you  will  be  ready  with  the  objection  that  if  the  potential 


114     THE  MOLECULAR  THEORY  OF  MATTER. 

energy  diminishes,  the  kinetic  energy  must  increase  ;  and  that 
this  would  imply  that  solution  is  always  accompanied  byya 
rise  in  temperature,  which  is  by  no  means  the  fact.  You  must 
bear  in  mind,  however,  that  the  properties  of  solids  show 
that  the  motions  of  the  molecules  of  these  bodies  are 
restricted  in  some  way,  and  that  the  process  of  solution 
undoubtedly  removes  these  restrictions,  so '  that  when  the 
molecules  of  the  solid  mingle  with  those  of  the  liquid,  they 
acquire  a  certain  additional  number  of  "  degrees  of  freedom." 
Hence,  although  the  diminution  of  potential  energy  certainly 
does  imply  an  increase  in  the  total  kinetic  energy,  it  does  not 
by  any  means  imply  an  increase  in  the  kinetic  energy  of 
translation.  That  is,  it  does  not  of  necessity  imply  an  increase 
in  temperature. 

Diffusion.  —  We  may  apply  this  same  principle  of  minimum 
potential  energy  in  discussing  the  diffusion  of  a  dissolved 
substance  through  the  solvent.  Suppose,  for  example,  that 
we  have  a  solution  in  which  the  concentration  is  different  in 
different  regions ;  then  the  potential  energy  of  the  system, 
per  unit  volume,  must  also  be  different  in  different  regions ; 
and  hence  the  molecules  of  the  dissolved  solid  will  tend 
towards  the  more  dilute  parts  or  towards  the  more  concen- 
trated parts,  according  as  the  potential  energy  of  the  system 
would  be  lessened  by  the  one  process  or  by  the  other.  In 
fact  it  can  be  easily  shown,  by  mathematical  reasoning,  that 
if  the  temperature  of  the  solution  be  maintained  constant  at 
all  points  (so  as  to  avoid  thermodynamical  considerations), 
the  potential  energy  of  the  system  will  necessarily  be  least 
when  the  concentration  is  everywhere  the  same  —  that  is, 
when  the  solution  is  homogeneous.  To  prove  this,  let  us 
assume  that  at  the  outset  the  solution  proposed  for  considera- 
tion has  different  degrees  of  concentration  in  different  parts, 
and  let  us  conceive  it  to  be  divided  by  imaginary  surfaces  into 
a  great  number  of  elementary  portions,  whose  volumes  are  vl} 
vzj  vs,  . . .  Furthermore,  let  us  represent  the  quantity  of  dis- 


DIFFUSION.  115 

solved  substance,  per  unit  volume,  in  these  respective  elemen- 
tary portions,  by  cb  c2,  cs,  .  .  .  ,  and  the  corresponding  potential 
energies  by  plt  p2)  ps,  ...  Then,  representing  the  total  quantity 
of  solid  in  solution  by  Q,  and  the  total  potential  energy  of  the 
system  by  P,  we  have 

Q  =  Wi  +  c2v2  +  csvs  +...  (42) 

and  P  =p1v1  +  p2v2  +  psvs  +  .  .  .  (43) 

We  are  required  to  discover  the  relations  that  must  exist 
among  the  c's,  in  order  that  P  shall  be  as  small  as  it  can  be, 
consistently  with  the  condition  (42).  It  is  shown,  in  works 
on  the  differential  calculus,  that  such  problems  can  be  solved 
by  differentiating  the  expression 

P-kQ  (44) 

with  respect  to  each  of  the  variables  <?1?  c2,  c3,  .  .  .  ,  and  equating 
all  the  resulting  differential  coefficients  to  zero  ;  k  being  a 
multiplier  at  present  undetermined,  and  which,  although  it 
may  not  be  a  constant,  is  nevertheless  to  be  treated  as  such  in 
performing  the  differentiations.  The  expression  (44)  is 
equivalent  to 


and  upon  differentiating  this  successively  with  regard  to 
«a,  c3,  .  .  .  ,  and  equating  the  coefficients  to  zero,  we  have 


or,  which  is  the  same  thing, 

^!_£  =  0,  f-*-k  =  0,  f*-ft  =  0,...  (45) 

dcl  dc2  des 

These  are  the  conditions  that  must  subsist  in  order  that  the 
potential  energy  of  the  system  may  have  its  smallest  possible 
value;  and  by  interpreting  them  we  must  discover  what 
relations  exist  among  the  c's.  Let  us  assume  that 


116      THE  MOLECULAR  THEORY  OF  MATTER. 

Then  (45)  is  equivalent  to 

f'(cl)=f'(c2)=f(cs)  =  ...  =  k.  (46) 

If  this  relation  were  a  general  one,  we  might  conclude  from 
it  that 


p  =  j  Me ; 


but  it  must  be  borne  in  mind  that  equations  (45)  are  true 
only  when  the  solution  under  consideration  has  reached  its  final 
condition  of  equilibrium.  Hence  we  can  infer  nothing  with 
regard  to  the  form  of  the  function  f(c),  and  we  are  driven  to 
the  conclusion  that  (46)  is  true,  when  the  final  state  of 
equilibrium  has  been  attained,  regardless  of  the  form  of  /(c). 
But  that  is  impossible  unless 


that  is,  unless  the  concentration  of  the  solution  is  everywhere 
the  same.  There  is  one  point  about  out  analysis  that  ought 
to  be  a  little  more  carefully  considered.  The  differential 
coefficients  of  (44)  would  be  equal  to  zero  whether  P  were  a 
minimum  or  a  maximum;  and  hence  our  reasoning  contains  a 
mathematical  ambiguity  which  must  be  further  examined.  If 
P  were  a  maximum  when  the  concentration  is  everywhere  the 
same,  it  follows  that  diffusion  would  increase  the  differences 
of  concentration  that  existed  in  the  original  solution,  until 
finally  the  concentration  would  become  zero  in  some  places, 
and  as  great,  in  other  places,  as  the  nature  of  the  dissolved 
substance  would  allow.  In  other  words,  the  dissolved 
substance  would  be  entirely  precipitated  from  solution.  The 
general  conclusion  to  which  our  mathematical  reasoning  leads 
therefore  is  that  if  a  given  solid  dissolves  in  a  given  liquid, 
its  solution  will  ultimately  become  homogeneous  ;  while  if  it 
does  not  dissolve,  the  reason  is  that  the  potential  energy  of 
the  system  is  less  when  the  solid  is  undissolved  than  it  would 
be  if  it  were  dissolved. 


OSMOTIC   PRESSURE. 


117 


Osmotic  Pressure.  —  Passing  over  the  earlier  experiments 
on  this  subject,  I  shall  proceed  at  once  to  those  made  by 
Pfeffer.  His  apparatus  is  shown  in  Fig.  40.*  It  consists 
essentially  of  a  clay  cell,  Z,  whose  pores  have  been  closed  by 
a  precipitate  of  ferrocyanide  of  copper.  A  cell  thus  prepared 
allows  water  to  pass  through  it  freely,  but  is  impervious  to 
such  solids  as  may  be  dissolved  in  the 
water.  In  using  the  apparatus  the  cell, 
Z,  is  filled  with  a  solution  of  some 
substance  —  say  of  sugar  or  nitre  —  of 
known  strength,  and  is  then  immersed 
in  water.  It  is  found,  under  these  cir- 
cumstances, that  water  passes  from  the 
containing  vessel  into  the  clay  cell, 
thereby  giving  rise  to  a  pressure  called 
the  "osmotic  pressure,"  which  can  be 
read  off  by  the  mercury  manometer,  A, 
and  which  is  often  surprisingly  great. 
To  understand  this  phenomenon  we 
have  to  fall  back  once  more  on  our 
fundamental  principle,  that  when  the 
system  is  in  equilibrium  its  potential 
energy  is  a  minimum.  As  I  told  you  a 
moment  ago,  the  potential  energy  of  .a 
solution  whose  temperature  is  every- 
where the  same  is  least  when  the 
dissolved  substance  is  uniformly  dis- 
tributed ;  and  it  follows  that  the  total 
potential  energy  of  the  liquid  system  in 

Pfeffer's  apparatus  would  be  lessened  if  the  dissolved  solid 
within  the  clay  cell  should  diffuse  outward  into  the  water  in 
the  containing  vessel.  It  cannot  diffuse  outward,  however,  for 

*  The  cut  here  presented  is  taken  from  that  given  on  page  98  of 
Ostwald's  Solutions,  to  which  book  the  reader  is  referred  for  further 
particulars  concerning  Pfeffer's  experiments,  and  for  detailed  information 
with  regard  to  the  preparation  of  the  clay  cells. 


FIG.  40.  —  PFEFFEB'S 
APPARATUS. 


118  THE   MOLECULAR   THEORY   OF   MATTER. 

the  cell  has  purposely  been  so  prepared  that  it  will  allow 
nothing  to  pass  but  pure  water.  Hence  if  the  dissolved  sub- 
stance is  to  become  disseminated  uniformly  throughout  the 
entire  mass  of  water,  it  will  have  to  do  so  by  some  other 
process.  The  only  other  process  possible  is  by  the  water  in 
the  outer  vessel  entering  the  cell;  and,  as  I  have  already  told 
you,  that  is  what  actually  happens.  It  is  another  case  of 
Mahomet  and  the  mountain.  Water  will  not  pass  into  the 
clay  cell  indefinitely,  however,  for  as  soon  as  the  pressure 
becomes  greater  inside  than  it  is  outside  (which  happens  at 
the  very  beginning  of  the  process),  energy  has  to  be  ex- 
pended, or  stored  up,  in  forcing  more  water  in,  against 
this  pressure.  The  total  energy  thus  expended  is  equal 
to  the  work  done  in  raising  the  mercury  column  in  the 
manometer ;  and  water  will  cease  to  enter  the  cell  the 
moment  that  the  energy  stored  up  in  the  mercury  column, 
owing  to  the  entrance  of  a  small  mass  of  water,  becomes 
equal  to  the  diminution  in  the  potential  energy  of  the 
system  due  to  the  entry  of  this  water.  Hence  in  every  case 
there  is  a  limiting  value  of  the  osmotic  pressure,  depending 
upon  the  concentration  of  the  solution  and  on  the  nature 
of  the  dissolved  substance ;  and  this  limiting  value  is  usually 
understood  to  be  meant  when  the  term  "  osmotic  pressure  "  is 
used  without  further  qualification.  Pfeffer  found  that  a  6  per 
cent  solution  of  cane  sugar  gave  an  osmotic  pressure  of  307.5 
centimeters  of  mercury,  or  over  four  atmospheres.  With  a 
3.3  per  cent  solution  of  nitre  he  obtained  a  pressure  of  436.8 
centimeters,  or  about  5f  atmospheres.  Pfeffer  also  found 
that  the  osmotic  pressure  is  proportional  to  the  concentration 
of  the  solution,  and  Van't  Hoff  has  further  shown,  by  thermo- 
dynamical  reasoning,  that  it  is  proportional  to  the  absolute 
temperature.  Hence,  if  p  is  the  osmotic  pressure  and  v  is  the 
number  of  cubic  centimeters  of  solution  that  contain  one 
gramme  of  the  dissolved  substance,  we  have 

p=^,  or  pv  =  fir,  (47) 


OSMOTIC    PRESSURE.  119 

which  is  entirely  analogous  to  equation  (27),  previously 
established  for  gases.  It  will  be  instructive  to  examine  (47) 
a  little  more  closely.  Suppose,  for  the  moment,  that  sugar 
could  exist  in  the  form  of  a  gas,  without  losing  its  essential 
properties  or  undergoing  any  modification  in  molecular  weight. 
Then,  since  the  molecular  weight  of  sugar  is  342,  we  know 
that  the  density  of  the  sugar-gas  would  be  171  times  the 
density  that  hydrogen  would  have  under  similar  circumstances 
of  temperature  and  pressure.  Now  the  volume  of  a  gramme 
of  hydrogen,  at  atmospheric  pressure  and  O0!).,  is  11,158  cubic 
centimeters  ;  and  hence  the  volume  of  a  gramme  of  sugar-gas, 
under  the  same  conditions,  would  be  11,158  -f-  171  =  65.25 
cubic  centimeters.  Substituting  this  for  v  in  the  gas-equation 
(27),  and  putting  r  —  273°  and  p  —  1033  grammes  per  square 
centimeter,  we  find  that  for  sugar-gas  the  constant  R  would 


Keturning  to  equation  (47)  let  us  find  the  value  of  R  that 
applies  when  p  represents  the  osmotic  pressure  of  sugar.  For 
a  one  per  cent  solution,  at  0°  C.,  Pf  effer  found  that  p  =  671 
grammes  per  square  centimeter.  Hence  putting  v  =  100  and 
p  =  671,  we  have,  from  (47), 


pv 
~~~ 


671  X  100 
~273" 


which  is  substantially  the  same  as  the  value  we  obtained  for 
sugar-gas.  Ostwald  expresses  this  fact  in  the  following 
words  :  "  The  osmotic  pressure  of  a  sugar  solution  has  the 
same  value  as  the  pressure  that  the  sugar  would  exert,  if  it 
were  contained  as  a  gas  in  the  volume  that  is  occupied  by  the 
solution.  The  gas  equation,  pv  —  RT,  holds  unchanged,  with 
the  same  constant,  for  solutions  ;  only  that  p  here  denotes  the 
osmotic  pressure."  This  important  and  suggestive  principle 
was  discovered  by  Van't  Hoff.  It  applies  to  many  substances 


120      THE  MOLECULAR  THEORY  OF  MATTER. 

besides  sugar,  and  to  many  other  substances  it  does  not  apply 
at  all.  I  cannot  digress  upon  the  conditions  of  its  applica- 
bility, but  must  refer  you,  for  further  information,  to 
Ostwald's  Outlines  of  General  Chemistry,  and  to  his  Solutions. 
Personally,  I  am  not  satisfied  that  gaseous  pressure  and 
osmotic  pressure  are  more  than  analogous  with  each  other, 
and  I  strongly  question  the  propriety  of  saying  that  "the 
state  of  substances  in  solution  is  in  the  widest  sense  com- 
parable "  with  the  gaseous  state. 

Electrolysis.  —  The  accepted  theory  concerning  electrolytic 
action  in  liquids  and  solutions  was  stated  by  Maxwell  with 
such  simplicity  that  I  will  read  you  what  he  says  :  "  A  very 
interesting  part  of  molecular  science,  which  has  not  been 
thoroughly  worked  out,  ...  is  the  theory  of  electrolysis. 
Here  an  electromotive  force  acting  on  a  liquid  electrolyte 
causes  the  molecules  of  one  of  its  components  to  be  urged  in 
one  direction,  while  those  of  the  other  component  are  urged 
in  the  opposite  direction.  Now  these  components  are  joined 
together  in  pairs  by  chemical  forces  of  great  power,  so  that 
we  might  expect  that  no  electrolytic,  effect  could  take  place 
unless  the  electromotive  force  was  so  strong  as  to  be  able  to 
tear  these  couples  asunder.  But,  according  to  Clausius,  in 
the  dance  of  molecules  which  is  always  going  on,  some  of  the 
linked  pairs  of  molecules  acquire  such  velocities  that  when 
they  have  an  encounter  with  a  pair  also  in  violent  motion  the 
molecules  composing  one  or  both  of  the  pairs  are  torn  asunder, 
and  wander  about  seeking  new  partners.  If  the  temperature 
is  so  high  that  the  general  agitation  is  so  violent  that  more 
pairs  of  molecules  are  torn  asunder  than  can  pair  again  in  an 
equal  time,  we  have  the  phenomenon  of  dissociation,  studied 
by  M.  St.-Claire  Deville.  If,  on  the  other  hand,  the  separated 
molecules  can  always  find  partners  before  they  are  rejected 
from  the  system,  the  composition  of  the  system  remains 
apparently  the  same.  Now  Professor  Clausius  considers  that 
it  is  during  these  temporary  separations  that  the  electromotive 


SATURATION.  121 

force  comes  into  play  as  a  directing  power,  causing  the  mole- 
cules of  one  component  to  move,  on  the  whole,  one  way,  and 
those  of  the  other,  the  opposite  way.  Thus  the  component 
molecules  are  always  changing  partners,  even  when  no  electro- 
motive force  is  in  action,  and  the  only  effect  of  this  force  is 
to  give  direction  to  those  movements  which  are  already  going 
on."* 

Saturation.  —  Thus  far  we  have  said  nothing  about  the 
quantity  of  solid  that  can  be  dissolved  in  a  given  mass  of 
liquid.  In  some  cases  it  appears  that  the  solvent  is  capable 
of  taking  up  an  almost  indefinite  amount  of  solid,  the  result- 
ing "  solution "  varying  from  a  moist  solid  to  an  undoubted 
liquid ;  but  in  most  cases  the  solvent,  at  any  given  tempera- 
ture, can  take  up  only  a  definite  quantity  of  solid.  A  solution 
that  contains  as  much  of  any  given  substance  as  it  is  capable 
of  dissolving,  is  said  to  be  saturated  with  respect  to  that 
substance.  If  to  such  a  saturated  solution  we  add  more  of 
the  solid,  we  can  see  no  change.  The  solid  that  we  have 
added  does  not  lessen,  nor  does  it  increase.  Such  a  system  is 
closely  analogous  to  a  closed  vessel  containing  a  liquid  in  con- 
tact with  its  saturated  vapor ;  and  the  explanation  of  the 
apparent  equilibrium  is  substantially  the  same  in  both  cases. 
The  solid  does  lose  molecules  from  its  surface,  but  an  equal 
number  return  to  it  from  the  liquid ;  and  hence  the  apparent 
equilibrium  and  quiescence.  If  the  system  is  disturbed  by 
altering  the  temperature,  or  by  evaporating  some  of  the  liquid 
and  thus  increasing  the  concentration,  the  equality  of  the 
exchanges  that  are  taking  place  at  the  surface  of  the  sub- 
merged solid  is  destroyed,  and  the  concentration  of  the 
solution  increases  or  diminishes  until  a  new  point  of  equilib- 
rium is  reached,  at  which  the  number  of  molecules  leaving  the 
solid  in  a  unit  time  again  becomes  equal  to  the  number  of 
those  returning  to  it.  By  the  same  kind  of  reasoning  that 
was  employed  in  considering  evaporation  from  a  liquid,  we 

*  Maxwell,  Theory  of  Heat  (ninth  edition,  1888),  page  325. 


122      THE  MOLECULAR  THEORY  OF  MATTER. 

can  easily  show  that  the  equilibrium  between  a  solid  and  its 
saturated  solution  is  entirely  independent  both  of  the  quantity 
of  the  undissolved  solid,  and  of  the  area  of  its  exposed  surface. 

Distillation.  —  One  of  the  most  striking  things  about  a 
solution  is,  that  if  it  is  allowed  to  evaporate,  the  solvent  passes 
off  alone,  leaving  behind  it  the  solid  that  was  previously  dis- 
solved. This  circumstance,  which  is  of  extraordinary  value 
in  the  arts,  seems  to  be  passed  over  without  adequate  dis- 
cussion by  writers  on  the  theory  of  solutions.  If  the  mole- 
cules of  the  dissolved  solid  are  moving  freely  about  among 
the  molecules  of  the  solvent,  we  should  naturally  expect  them 
to  escape  from  the  surface  of  the  solution  with  a  frequency 
not  altogether  insensible  in  comparison  with  the  frequency  of 
escape  of  the  solvent-molecules;  but  this  is  not  the  fact. 
If,  as  the  advocates  of  the  "hydrate  theory"  of  solutions 
believe,  the  dissolved  substance  is  combined  with  the  solvent 
in  some  manner,  so  that  its  molecules  are  relatively  massive,  — 
and  if,  furthermore,  there  is,  for  liquids,  some  proposition 
analogous  to  Boltzmann's  law  of  distribution  of  kinetic  energy 
in  gases, — 'then  it  might  follow  that  no  sensible  proportion 
of  the  ponderous,  hydrated  molecules  of  the  dissolved  sub- 
stance can  ever  move  fast  enough  to  overcome  the  attraction 
of  the  liquid,  and  escape  from  it  by  the  process  of  evaporation. 
On  the  other  hand,  if  any  considerable  proportion  of  the 
molecules  of  a  dissolved  electrolyte  are  simultaneously  in  a 
state  of  dissociation  (as  maintained  by  the  advocates  of  the 
"  physical  theory  "  of  solutions),  it  is  not  easy  to  see  how  an 
adequate,  mechanical  explanation  of  the  facts  of  distillation 
can  be  given.  I  make  this  statement  solely  as  a  confession  of 
my  own  ignorance,  and  the  advocates  of  the  "  hydrate  theory  " 
of  solutions  are  welcome  to  such  comfort  as  they  can  extract 
from  it.  Of  course  we  can  always  fall  back  on  the  fact  that 
most  solids  are  non-volatile  at  temperatures  to  which  their 
solutions  are  ordinarily  submitted.  This,  together  with  the 
further  fact  that  solution  implies  a  fall  in  potential  energy, 


SUPERS  ATUKATION.  123 

might  be  considered  as  affording  a  basis  for  explaining  the 
fixity  of  dissolved  substances ;  for  if  the  external  molecules 
of  the  solid  were  originally  held  in  place  by  the  attraction  of 
the  solid,  and  if  the  attraction  of  the  liquid  for  these  mole- 
cules is  greater  than  the  attraction  of  the  solid  for  them,  then 
one  might  possibly  infer  that  the  non-volatility  of  the  original 
solid  implies  its  non-vaporization  from  solution.  But  I 
strongly  doubt  if  this  argument  would  commend  itself  to  one 
who  did  not  know  the  facts  in  advance.  Moreover,  it  is  not  a 
mechanical  explanation  at  all,  in  the  sense  in  which  I  under- 
stand the  phrase. 

Super  saturation.  —  If  we  cool  a  saturated  solution  of  a 
substance  that  is  more  soluble  at  higher  temperatures  than  at 
lower  ones,  under  ordinary  circumstances  there  will  be  a 
deposition  of  the  solid.  If,  however,  care  is  taken  to  exclude 
all  traces  of  free  solid  from  the  solution,  the  cooling  may 
often  be  carried  to  a  point  considerably  below  the  temperature 
normally  corresponding  to  the  given  degree  of  saturation, 
without  deposition  taking  place.  Solutions  in  this  condition 
are  said  to  be  supersaturated.  If  I  have  made  myself  intelli- 
gible in  explaining  the  phenomena  of  saturation,  you  will 
have  no  difficulty,  I  think,  in  understanding  supersaturation. 
The  concentration  of  a  solution  containing  'an  excess  of  solid 
is  determined  by  the  equality  of  the  molecular  exchanges  that 
take  place  at  the  surface  of  the  solid ;  and  hence  it  is  plain 
that  when  no  such  free  solid  is  present,  the  cause  that 
normally  determines  the  degree  of  concentration  is  also  not 
present,  —  and,  in  fact,  the  conception  of  a  definite  point  of 
"  saturation "  is  no  longer  applicable.  If  a  particle  of  the 
dissolved  solid,  or  of  a  substance  isomorphous  with  it,  be 
placed  in  a  supersaturated  solution,  the  dissolved  substance  is 
rapidly  deposited  about  the  submerged  particle  as  a  nucleus, 
until  the  concentration  of  the  solution  is  reduced  to  its 
normal  value.  A  solution  cannot  be  cooled  indefinitely,  with- 
out deposition,  even  when  all  free  solid  is  excluded  with  the 


124      THE  MOLECULAR  THEORY  OF  MATTER. 

greatest  care.  The  reason  for  this  seems  to  be,  that  in  the 
ceaseless  re-arrangement  of  molecules  that  takes  place  in  a 
liquid,  it  occasionally  happens  that  a  certain  number  of  mole- 
cules of  the  dissolved  substance  fortuitously  come  together  in 
such  a  way  as  to  serve  as  a  nucleus  for  the  deposition  of  the 
solid;  and  when  this  occurs,  the  supersaturated  solution 
spontaneously  deposits  the  excess  of  solid  that  it  contains. 

Crystals.  —  The  fundamental  fact  of  crystallization  is  thus 
lucidly  stated  by  Williams  *  :  "All  chemically  homogeneous 
substances,  when  they  solidify  from  a  state  of  vapor,  fusion, 
or  solution,  tend  to  assume  certain  regular  polyhedral  forms. 
This  tendency  is  much  stronger  in  some  substances  than  in 
others,  and  it  varies  widely  in  the  same  substance  under 
different  physical  conditions.  The  regularly  bounded  forms 
thus  assumed  by  solidifying  substances  are  called  crystals" 
Crystals  of  different  substances  have  different  shapes,  and 
sometimes  the  same  substance  is  deposited  in  different  forms 
under  different  conditions.  But  it  is  not  alone  in  the  regu- 
larity of  their  shape  that  crystals  are  peculiar.  If  we  examine 
these  bodies  we  find  that,  in  general,  they  exhibit  different 
properties  in  different  directions.  Homogeneous,  uncrystal- 
lized  bodies,  such  as  glass,  exhibit  the  same  cohesion,  the 
same  hardness,  the  same  elasticity,  and  the  same  thermal, 
optical,  and  electrical  properties,  wherever  we  examine  them  ; 
but  in  crystals  it  is  found  that,  with  certain  exceptions,  these 
properties  are  the  same  along  directions  that  are  parallel,  but 
different  along  directions  that  are  not  parallel.  In  other 
words,  homogeneous,  uncrystallized  bodies  are  isotropic;  while 
crystals,  generally  speaking,  are  not  isotropic.  These  points 
are  described  in  all  the  works  on  crystallography,  but  crystals 
also  exhibit  many  other  interesting  properties  which  are 
seldom  discussed  in  such  books  or  journals  as  are  accessible 
to  the  general  public.  For  example,  if  a  crystal  be  removed 
from  a  solution  in  which  it  is  forming,  and  be  carefully  pre- 

*  Elements  of  Crystallography,  page  1. 


BOUNDING   PLANES    OF    CRYSTALS.  125 

served,  it  never  loses  the  power  of  resuming  its  growth.  If 
at  any  future  time  it  is  submerged  in  a  solution  similar  to 
that  in  which  it  was  first  formed,  the  invisible  forces  again 
assert  themselves,  and  the  crystal  slowly  enlarges,  precisely 
as  if  there  had  been  no  interruption.  A  crystal  may  even 
have  been  produced  in  some  former  geological  period,  thou- 
sands of  centuries  ago ;  and  yet,  upon  placing  it  in  a  suitable 
solution,  we  find  that  the  work  of  molecular  architecture  is  at 
once  resumed,  just  as  though  all  those  intermediate  ages  were 
blotted  out.  The  crystal  may  even  be  almost  entirely  de- 
stroyed in  the  interval  of  inactivity,  and  yet  it  will  grow  as 
before,  provided  there  remains  within  it  some  small  fragment 
that  has  the  structure  of  the  primitive  crystal.  Crystals  also 
possess  the  power  of  self-repair  to  a  considerable  extent,  so 
that  if  they  are  scored  or  bruised  the  subsequent  growth  is 
abnormally  rapid  over  the  injured  areas,  until  the  injuries 
disappear  and  the  crystal  regains  its  perfect  form.* 

Bounding  Planes  of  Crystals.  —  In  order  to  facilitate  the 
discussion  of  the  planes  that  form  the  limiting  surfaces  of  a 
crystal,  let  us  conceive  axes  to  be  drawn  within  the  crystal, 
and  let  us  refer  the  bounding  planes  to  these  axes,  precisely 
as  we  do  in  the  study  of  analytical  geometry  of  three  dimen- 
sions. The  positions  of  the  axes  are  to  be  determined  by 
considering  the  symmetry  of  the  crystal ;  but  as  you  have  no 
doubt  made  some  study  of  crystals  in  connection  with  your 
courses  in  geology  and  mineralogy,  I  shall  not  dwell  upon  the 
assignment  of  axes  in  the  various  systems  of  crystals  that 
occur  in  nature,  but  shall  assume  that  they  are  already  drawn. 
Let  us  therefore  pass  to  the  consideration  of  some  actual 
crystal  —  for  example,  the  typical  octahedral  form  shown  in 
Fig.  41.  This  crystal  is  perfect  —  that  is,  it  is  symmetrical 
with  respect  to  each  of  the  three  axes  (shown  by  the  dotted 
lines),  and  it  is  not  modified  by  the  loss  of  its  edges,  nor  in 

*  See  Professor  John  W.  Judd's  excellent  discourse,  The  Rejuvenescence 
of  Crystals,  in  Nature  for  May  28,  1891. 


126 


THE  MOLECULAR  THEORY  OF  MATTER. 


any  other  way.  Fig.  42  is  an  enlarged  view  of  the  shaded 
face  of  Fig.  41,  and  with  the  notation  given  in  the  figure  it  is 
plain  that  the  equation  of  this  face  is 

-  +  7  +  -  =  l.  (48) 

a      o       c 

It  is  also  evident  that  all  the  other  faces  of  this  crystal  are 
obtainable  from  (48)  by  changing  the  sign  of  one  or  more  of 
the  quantities  a,  b,  and  c.  A  similar  equation  can  be  found 
for  each  face  of  any  given  crystal  whatever ;  and  observation 


a 

FIGS.  41  and  42.  —  ILLUSTRATING  THE  GEOMETRY  OF  A  CRYSTAL. 

indicates  the  very  remarkable  fact  that  in  all  planes  that 
actually  occur  in  crystals,  the  quantities  a,  b,  c,  either  bear  a 
simple  ratio  to  one  another,  or  are  infinite.  For  example,  if 
the  equation  of  any  given  crystal-face  were  found,  we  might 
have 


a:b  :  c  = 


:2,  or  a  :  &  :  c  =  4  :  5  :  6,  or  a  :  b  :  c  =  2  :  3  :oo  , 

or  some  other  similar  proportion  ;  but  we  should  never  find 
any  crystal  plane  in  which  the  ratio  was  anything  like"  this  : 

a:b:c  =  l:7:  Vl3. 

This  law  —  called  the  "  law  of  rationality  of  the  intercepts  " 
(or  indices)  —  like  many  another  one  in  the  domain  of 
physics,  is  seldom  fulfilled  with  strict  accuracy;  and  yet  it 


MOLECULAR  STRUCTURE  OF  CRYSTALS.      127 

comes  so  near  the  truth  that  we  must  regard  it  as  the  visible 
expression  of  some  structural  simplicity  within  the  crystal.* 

Molecular  Structure  of  Crystals.  —  The  molecular  structure 
of  crystals  has  been  investigated  by  Sohncke  and  others  from 
a  geometrical  standpoint,!  but  I  prefer  to  give  you  what  may 
be  called  a  "physical"  presentation  of  the  subject,  following 
Professor  Liveing  for  the  most  part.  $  Very  little  is  known 
about  the  constitution  of  solid  bodies  generally,  but  it  is  quite 
certain  that  in  crystals  there  is  some  regularity  of  orientation, 
either  in  the  molecules  themselves  or  in  their  motions.  The 
physical  properties  of  these  bodies  seem  to  prove  that  much, 
beyond  a  reasonable  doubt.  We  shall  take  it  for  granted  that 
this  regularity  is  of  such  a  nature  that  any  given  molecule,  in 
its  vibratory  excursions,  never  passes  outside  of  a  certain 
imaginary  ellipsoid  which  we  may  conceive  to  be  described 
about  the  mean  position  of  the  molecule.  Crystals  may  then 
be  regarded  as  aggregates  of  such  ellipsoids,  piled  up  in  such 
a  way  that  the  corresponding  axes  of  all  of  them  are  either 
parallel  throughout  the  mass,  or  at  least  arranged  in  accord- 
ance with  some  definite  geometrical  scheme.  If  we  now  apply 
our  general  principle,  that  the  potential  energy  of  a  molecular 
system  tends  towards  a  minimum,  we  see  that  when  a  sub- 
stance solidifies,  either  from  solution  or  from  a  state  of  fusion, 
the  ellipsoids  that  bound  its  molecules  must  take  such  posi- 
tions that  the  potential  energy  of  the  resulting  solid  is  as 
small  as  possible.  I  shall  assume  that  the  constitution  and 
mode  of  motion  of  the  molecules  under  consideration  are  such 
that  the  potential  energy  of  the  system  is  least  when  the 

*  For  a  more  accurate  form  of  the  law  of  rationality  of  the  indices,  see 
the  article  on  Crystallization  in  Watts's  Dictionary  of  Chemistry.  See 
also  Williams's  Crystallography. 

t  See  the  chapter  on  Crystals  in  Ostwald's  Outlines  of  General  Chem- 
istry, where  the  geometrical  theory  is  very  clearly  set  forth. 

|  Nature,  June  18,  1891,  page  156.  I  am  hardly  prepared  to  follow 
Professor  Liveing  in  extending  the  conception  of  "surface  tension"  to 
solids,  however. 


128      THE  MOLECULAR  THEORY  OF  MATTER. 

ellipsoids  are  piled  together  as  closely  as  possible ;  though,  as 
I  have  already  told  you,  this  is  by  no  means  a  necessary 
assumption,  nor  is  it,  speaking  generally,  even  a  probable  one. 
Moreover,  as  we  are  touching  upon  this  subject  for  the  sole 
purpose  of  showing  how  the  molecular  theory  explains  crystal 
forms,  and  without  any  intention  of  discussing  the  constitu- 
tion of  these  bodies  exhaustively,  I  shall  not  treat  of  the 
general  case  in  which  the  axes  of  the  ellipsoids  are  unequal, 
but  shall  confine  myself  to  that  special  one  in  which  they  are 
equal,  and  the  ellipsoids  are  spheres.  As  Professor  Liveing 
remarks,  "  the  problem  is  then  reduced  to  finding  how  to  pack 
the  greatest  number  of  equal  spherical  balls  into  a  given 
space."  You  will  find  it  interesting  and  instructive  to  work 
out  this  problem,  during  some  leisure  hour,  with  the  help  of 
a  liberal  supply  of  buck-shot  or  bullets.  Three  possible  solu- 
tions of  it  will  readily  occur  to  you.  In  the  first  place,  we 
can  arrange  a  layer  of  spheres  as  shown  in  Fig.  43,  where 
each  sphere  touches  four  others  ;  and  upon  this  layer  we  can 
place  another  one  in  the  manner  illustrated  by  the  four 


FIGS.  43  and  44.  — SQUARE  PYRAMID  OF  SPHERES. 

lighter  spheres  in  the  figure  —  each  sphere  in  the  second  layer 
coming  directly  over  one  of  the  interstices  in  the  first  one. 
By  adding  other  layers  in  the  same  way,  we  shall  eventually 
form  the  pyramid  shown  in  Fig.  44.  A  second  method  of 


MOLECULAR    STRUCTURE 

arranging  the  spheres  of  the  first  layer  is  shown  in  Fig.  45, 
where  each  of  the  interior  spheres  touches  six  others.  If  a 
second  layer  be  superposed  upon  this  first  one  in  the  manner 
indicated  by  the  three  light  spheres,  and  the  piling  is  continued 
in  the  same  manner,  we  shall  eventually  arrive  at  the  form  of 


FIGS.  45  and  46.  —  TRIANGULAR  PYRAMID  OF  SPHERES. 

pyramid  shown  in  Fig.  46.  You  will  find  that  there  are  two 
ways  of  placing  the  second  layer  upon  the  fundamental  one 
shown  in  Fig.  45.  One  of  these  methods,  if  continued,  gives 
the  pyramid  shown  in  Fig.  46,  while  the  other  gives  a  pyra- 
mid of  greater  height.  Fig.  46  is  really  half  of  a  cube,  its 
apex  being  a  corner  of  the  cube.  The  other  pyramid  to  which 
I  have  referred  is  a  regular  tetrahedron,  formed  on  Fig.  45  as 
a  base.  Now  although  these  three  methods  of  arranging  the 
spheres  may  seem  to  you,  at  first  sight,  to  bo.  distinct,  a  little 
thought  (assisted  perhaps  by  your  experimental  pile  of 
bullets)  will  show  you  that  the  internal  structure  is  identical 
in  all  of  them.  The  spheres  in  any  one  of  the  slant  faces  of 
Fig.  44  are  arranged  precisely  like  those  in  Fig.  45 ;  and  the 
slant  faces  of  Fig.  46,  on  the  other  hand,  correspond  exactly 
to  the  configuration  shown  in  Fig.  43.  There  is  only  one 
"  closest  way  "  to  pack  the  spheres,  and  if  they  are  piled  in 
that  way,  all  the  pyramids  that  I  have  referred  to  can  be 
obtained  by  passing  suitable  planes  through  the  mass.  I 
must  now  refer  briefly  to  what  may  perhaps  be  called  the 
fundamental  fact  of  crystallization  —  the  fact,  namely,  that 


130      THE  MOLECULAR  THEORY  OF  MATTER. 

the  bounding  surfaces  of  crystals  are  planes.  When  a  crystal 
is  forming,  we  have  to  conceive  that  a  continuous  series  of 
exchanges  is  going  on,  all  over  its  surface.  Molecules  of  the 
dissolved  substance  are  caught  by  the  attraction  of  the  grow- 
ing crystal,  but,  on  the  other  hand,  molecules  of  the  crystal 
are  continually  passing  into  solution  again ;  and  the  gradual 
increase  in  size  of  the  crystal  is  due  to  the  fact  that  in  a  unit 
time  more  molecules  are  caught  by  it  than  are  lost  again. 
With  this  much  premised,  let  us  turn  our  attention  to  Fig. 
47,  which  represents  a  sphere-pyramid  (or  crystal)  similar  to 
that  in  Fig.  44,  except  that  it  has  a  partially  completed  layer 

on  one  face.  Let  us  see  what 
will  happen  as  the  deposition 
of  spheres  proceeds.  If  a 
sphere  should  lodge  against 
a  slant  face  of  Fig.  47,  it 
would  touch  three  spheres  of 
the  pyramid,  and  the  force 
of  attraction  exerted  upon  it 
FIG.  47.  -  SQUARE  PYRAMID  WITH  by  the  molecules  within  these 

PARTIAL  LAYER  OF  SPHERES  ON       spheres    Would    be    the    chief 
ONE  SIDE.  " 

cause  of  its  retention  as  part 

of  the  crystal.  Let  us  next  assume  that  a  sphere  lodges 
on  the  little  ledge  formed  by  the  top  of  the  partial  layer 
shown  on  the  right.  It  will  then  touch  five  spheres,  and 
will  be  held  in  position  by  the  attractions  of  the  five  cor- 
responding molecules.  You  will  readily  see,  I  think,  that 
as  the  molecules  of  the  crystal  are  torn  off,  in  the  process  of 
re-solution  that  is  ever  going  on,  those  that  have  adhered  to 
the  slant  face  of  the  crystal  will  be  torn  away  with  greater 
frequency  than  those  that  have  lodged  on  the  little  ledge.  It 
follows  that  whenever  a  few  molecules  happen  to  be  deposited 
in  juxtaposition  on  the  face  of  a  crystal,  the  growth  about  the 
edges  of  the  layer  thus  begun  will  be  far  more  rapid  than  the 
sporadic  growth  that  occurs  elsewhere,  and  the  layer  will  be 
quickly  extended  until  it  covers  the  entire  face  of  the  crystal. 


MOLECULAR  STRUCTURE  OF  CRYSTALS.      131 

Hence  if  the  faces  of  the  crystal  are  plane  at  any  one  moment 
of  its  growth,  they  will  always  remain  so ;  and  as  the  imagi- 
nary surfaces  that  enclose  the  group  of  four  or  more  spheres 
that  constitute  the  beginnings  of  a  crystal  can  be  regarded  as 
planes,  the  reason  for  the  flatness  of  the  faces  of  the  finished 
crystal  becomes  apparent.  If  you  will  consider  the  bottom  face 
of  Fig.  44  in  this  same  manner,  you  will  find  that  it  will  grow 
considerably  faster  than  the  slant  faces,  and  that  the  crystal 
will  tend  towards  the  octahedral  form  that  would  be  obtained 
by  placing  two  of  these  square  pyramids  base  to  base.  The 
self-repair  of  crystals  is  also  explicable  by  the  same  line  of 
reasoning.  We  do  not  yet  fully  understand  the  phenomena 
of  crystallization,  and  hence  we  are  by  no  means  completely 
prepared  to  explain  them.  The  same  substance  will  often 
crystallize  in  different  forms  under  different  conditions,  and 
I  suppose  we  must  conclude  from  this  that  the  tendency  of 
the  molecule-spheres  to  adhere  to  the  various  faces  of  the 
growing  crystal  is  not  altogether  independent  of  the  circum- 
stances under  which  the  deposition  takes  place.  You  will 
find,  by  once  more  referring  to  your  bullets,  that  the  planes 
that  I  have  shown  in  the  diagrams  are  by  no  means  the  only 
ones  that  can  be  used  to  cut  out  crystalline  forms  from  the 
closely-piled  spheres.  The  planes  that  I  have  shown  are  those 
that  have  the  simplest  relations  among  their  "indices,"  or 
intercepts  ;  and  you  will  find,  by  trying  other  planes,  that  the 
faces  of  the  pile  become  increasingly  irregular  as  the  ratios  of 
the  intercepts  of  these  planes  depart  further  from  those  ratios 
that  can  be  expressed  by  small  integers  or  simple  fractions. 
This  gives  us  a  suggestion  of  the  meaning  of  the  "law  of 
rationality  of  the  intercepts,"  but  I  cannot  develop  this  sug- 
gestion further  at  the  present  time.  There  is  one  other  point 
that  I  must  touch  upon,  before  leaving  this  fascinating  subject 
of  crystal  forms,  and  that  is,  the  existence  of  planes  of  easiest 
cleavage.  If  we  try  to  split  a  crystal,  the  crack,  once  started, 
will  naturally  tend  to  follow  the  direction  of  least  resistance ; 
and  this  direction  will  be  that  in  which  the  number  of  sphere- 


132  THE   MOLECULAR   THEORY  OF  MATTER. 

contacts,  per  unit  area  of  fractured  surface,  will  be  least. 
There  is  no  difficulty  in  investigating  this  number  of  contacts 
along  any  given  plane  —  that  is,  there  is  no  difficulty  except 
that  the  labor  is  sometimes  quite  considerable.  It  is  easy  to 
show,  for  example,  that  in  the  cubical  crystal  of  which  Fig. 
46  is  one  half,  the  number  of  sphere-contacts  that  must  be 
broken,  per  unit  area  of  fractured  surface,  is 

2V?  (49) 


when  the  fracture  is  parallel  to  the  base  of  Fig.  46,  and 

4 


(50) 


when  it  is  parallel  to  the  front  face  of  this  figure  (d  being  the 
diameter  of  a  sphere).  Now  (49)  is  less  than  (50),  and  hence 
the  crystal  can  be  broken  along  a  direction  parallel  to  the 
base  of  Fig.  46  easier  than  it  can  be  along  a  direction  parallel 
to  one  of  the  cubic  faces  of  the  crystal.  This  corresponds  to 
the  known  fact  that  in  cubical  crystals  the  cleavage  is  usually 
octahedral.  I  have  tried  to  give  you  the  barest  outline  of  the 
application  of  the  molecular  theory  to  crystalline  bodies,  and 
I  should  like  to  dwell  longer  upon  it,  were  it  not  for  the  fact 
that  the  subject  is  so  extensive  that  no  adequate  presentation 
of  it  could  be  given  in  any  reasonable  time.  There  is  abun- 
dant room  for  research  here,  too ;  for  the  crystal-theory  has 
by  no  means  attained  its  final  form.  We  have,  as  yet,  only 
some  very  general  notions  concerning  it ;  and  if  I  have  suc- 
ceeded in  showing  you  the  direction  in  which  future  research 
is  likely  to  lead  us,  I  have  accomplished  all  that  I  could  hope 
to,  in  the  short  time  at  my  disposal. 


MOLECULAR  MAGNITUDES.  133 


V.  MOLECULAR  MAGNITUDES. 

Preliminary  Remarks.  —  Before  entering  upon  a  discussion 
of  the  sizes  of  molecules,  or  of  the  range  of  molecular  forces, 
we  ought  to  have  a  clear  idea  of  what  is  meant  by  these  terms. 
Unfortunately  we  do  not  yet  know  enough  about  molecules 
even  to  define,  in  a  satisfactory  manner,  what  is  meant  by 
their  "  size."  If  they  were  hard,  smooth,  material  spheres, 
we  should  know  what  is  meant  by  the  word ;  but  if  they  have 
any  other  shape,  or  any  other  constitution,  its  meaning  is  not 
so  clear.  Even  if  molecules  were  hard  bodies  of  definite  form, 
we  ought  to  specify  which  one  of  their  dimensions  is  intended 
when  we  speak  of  their  "  size  " ;  but  in  the  present  state  of 
knowledge  this  is  utterly  impossible,  and  hence  it  is  also 
impossible  to  compute  the  dimensions  of  molecules  with  any 
considerable  approach  to  precision.  We  have  had  to  be  con- 
tented, thus  far,  with  determinations  of  the  general  order  of 
magnitude  which  must  be  assigned  to  these  bodies  ;  and  it  is 
in  this  sense  alone  that  I  shall  speak  of  the  "  size  of  mole- 
cules "  this  evening.  In  the  original  form  of  the  kinetic 
theory  of  gases  the  molecules  were  assumed  to  be  spherical, 
in  spite  of  the  fact  that  they  were  believed  to  have  other 
forms,  because  such  a  supposition  simplified  the  mathematical 
treatment  of  the  subject ;  and  in  our  attempts  to  discover  the 
sizes  of  the  molecules  we  shall  again  assume  them  to  be 
spherical,  for  the  same  reason,  and  with  the  same  justification. 
We  shall  treat  them  as  spheres,  having  a  diameter  which 
represents,  in  some  sense,  the  average  diameter  of  the  actual 
molecules.  With  this  much  premised,  let  us  proceed  to 
review  a  few  of  the  methods  that  have  been  proposed  for 
finding  molecular  magnitudes.. 

The  Electrical  Method  for  Finding  the  Aggregate  Volume 
of  Molecules.  —  It  has  been  suggested  that  the  aggregate 
volume  of  the  molecules  of  mercury  vapor  can  be  found  by 


134  THE   MOLECULAR   THEORY    OF   MATTER. 

the  following  electrical  method  :  *  There  is  some  reason  for 
believing  that  the  molecules  of  mercury  vapor  actually  are 
spherical  [see  page  52]  ;  and  if  we  further  assume  that  they 
are  capable  of  conducting  electricity,  it  can  be  shown  that 
the  "specific  inductive  capacity,"  K,  of  the  vapor,  can  be 
calculated  by  means  of  the  formula 


where  e  stands  for  the  ratio  which  the  aggregate  volume  of 
the  spherical  molecules  bears  to  the  total  space  in  which  they 
are  confined.  By  solving  this  equation  for  c  we  have 


from  which  we  could  compute  c  if  we  knew  K.  I  do  not  find 
that  the  specific  inductive  capacity  of  mercury  vapor  has  ever 
been  determined,  and  hence  I  cannot  show  you  the  numerical 
result  that  the  electrical  method  would  give.  Since  equation 
(51)  can  be  deduced  only  when  the  molecules  are  known  to 
be  conducting  spheres,  it  is  manifestly  unfair  to  apply  it  to 
such  gases  as  hydrogen  and  oxygen,  whose  molecules  are 
certainly  not  spherical.  If  we  knew  the  forms  of  the  mole- 
cules of  these  gases  it  might  be  possible  to  construct  some 
equation  analogous  to  (51),  which  would  be  applicable  to 
them  ;  but  in  the  absence  of  such  knowledge  it  does  not 
appear  profitable  to  dwell  longer  upon  the  electrical  method 
for  determining  the  aggregate  volume  of  molecules. 

Aggregate  Volume  of  Molecules,  from  the  Gas  Equation.  — 
If  we  return  to  the  corrected  gas  equation,  as  given  by 
Clausius  for  carbon  dioxide  (30),  and  consider  the  effect  of 
putting  v  =  b  in  that  equation,  we  shall  see  that  the  result 
would  be  to  make  p  =  oo  .  Now  we  cannot  imagine  that  the 
molecules  of  a  gas,  when  under  an  infinite  pressure,  could 
still  remain  at  any  finite  distance  from  one  another;  and 

*  See  Encyclopedia  Britannica,  article  Molecule,  page  620. 


REMARKS    ON   THE   FOREGOING   RESULTS.  135 

hence  we  infer  that  if  the  volume  of  a  given  mass  of  carbon 
dioxide  should  be  reduced,  by  pressure,  until  v  =  b,  the  mole- 
cules of  this  mass  of  gas  would  be  brought  into  absolute 
contact  with  one  another.  The  value  of  b,  as  given  by  Clau- 
sius,  is  .000843 ;  and  we  infer  from  this  that  the  total  space 
required  for  the  molecules  of  carbon  dioxide,  when  these 
molecules  are  in  contact  with  one  another,  is  .000843  of  the 
space  occupied  by  the  gas  at  the  freezing  point  of  water,  and 
under  atmospheric  pressure.  However,  this  space  is  not  all 
actually  occupied  by  the  molecules,  for  there  must  be  vacant 
interstices.  In  fact,  it  is  easily  shown  that  if  the  molecules 
are  spheres,  packed  together  in  the  closest  way  possible,  the 
actual  aggregate  volume  of  the  molecules  will  be  very  nearly 
f  of  their  apparent  volume ;  and  hence  we  conclude  that  in 
carbon  dioxide  the  actual  aggregate  volume  of  the  molecules 
is  f  X  .000843  =  .000723  of  the  space  occupied  by  the  gas 
itself  at  0°  C.  and  atmospheric  pressure.  "Corrected"  equa- 
tions have  been  given  for  other  gases  by  various  observers,  and 
from  these,  by  a  process  of  reasoning  precisely  like  that  I 
have  just  given  you,  the  aggregate  molecular  volumes  of  the 
corresponding  gases  may  be  obtained.  Thus  from  equations 
given  by  Mr.  William  Sutherland*  I  have  obtained  the 
following  values  of  the  aggregate  molecular  volumes  of  certain 
gases,  the  unit  in  each  case  being  the  volume  of  the  gas  it- 
self at  0°  C.  and  one  atmosphere  pressure.  For  H,  c  =  .00033 ; 
for  N,  €  =  .00087;  and  for  0,  e  =  .00074. 

Remarks  on  the  Foregoing  Results.  —  If  the  volume  of 
the  molecule  is  unaltered  when  a  substance  passes  from  the 
gaseous  to  the  liquid  and  solid  states,  it  is  evident  that  we 
have  a  sort  of  check  which  can  be  applied  to  the  results  we 
have  just  obtained,  so  far  as  the  available  experimental  data 
will  allow.  Thus,  if  we  should  compress  carbon  dioxide  until 
its  molecules  were  brought  into  absolute  contact  with  one 

*  See  his  paper  on  The  Laws  of  Molecular  Force,  in  the  Philosophical 
Magazine  for  March,  1893,  page  232. 


136 


THE  MOLECULAR  THEOEY  OF  MATTER. 


another,  the  volume  of  the  resulting  mass  would  be  .000843 
of  the  volume  occupied  by  the  gas  at  0°  C.  and  under 
atmospheric  pressure.  The  density  of  this  mass  would  there- 
fore be  equal  to  the  density  of  the  original  gas  divided  by 
.000843.  Now  Kegnault  found  that  at  0°  C.,  and  one  atmos- 
phere pressure,  a  cubic  decimeter  of  carbon  dioxide  weighs 
1.9774  grammes ;  and  therefore  the  weight  of  a  cubic  deci- 
meter of  carbon  dioxide  molecules,  when  in  actual  contact,  is 
1.9774  -f.  .000843  =  2.346  kilograms.  This  is  the  greatest 
density  carbon  dioxide  could  have,  in  any  form,  if  our 
calculations  are  correct,  and  if  the  molecules  of  this  gas 
preserve  their  bulk  when  the  gas  changes  its  state.  The 
greatest  observed  density  that  I  know  of,  for  this  gas,  is  1.6 
kilograms  per  cubic  decimeter.  This  value  was  observed  in 
carbon  dioxide  that  had  been  solidified  and  hammered.  The 
greatest  observed  density,  you  will  see,  is  only  about  two- 
thirds  of  the  maximum  possible  limit  as  given  by  our  com- 
putation. Similar  calculations  are  readily  made  for  the 
remaining  three  gases  for  which  we  have  computed  the  value 
of  e.  The  results  are  as  follows  : 

MAXIMUM  DENSITIES  OF  H,  N,  O,  AND  C02. 


GAS. 

NORMAL 
DENSITY.  * 

GREATEST  POS- 
SIBLE DENSITY 
(Calculated). 

GREATEST 
OBSERVED 
DENSITY. 

Hydrogen   .... 
Nitrogen     .... 
Oxygen  

0.0896  gms. 
1.2562 
1.4298 

0.232  kg. 
1.232 
1.653 

0.089  kg. 
0.905 
1.24 

Carbon  dioxide    .     . 

1.9774 

2.346 

1.60 

The  "  greatest  observed  densities"  here  quoted  are -from  the 
following  authorities  :  For  hydrogen  I  have  taken  the  density 
at  3,000  atmospheres,  as  given  by  Amagat's  experiments,  and 
quoted  in  Preston's  Theory  of  Heat;  for  nitrogen  I  have 

*  That  is,  the  "weight  of  a  cubic  decimeter,  in  grammes,  at  0°C.  and 
one  atmosphere  pressure."  The  densities  in  the  last  two  columns  of  the 
table  are  expressed  in  kilograms  per  cubic  decimeter. 


REMARKS    ON   THE   FOREGOING   RESULTS.  137 

taken  Olszewski's  greatest  result,  obtained  at  — 194°  C. ;  for 
oxygen  I  have  taken  Wroblevsky's  result,  obtained  at 
—  200°  C. ;  and  the  density  given  for  carbon  dioxide  is  due 
to  Dewar.  By  observing  the  increase  in  volume  of  palladium 
when  this  metal  is  caused  to  occlude  a  given  weight  of 
hydrogen,  Dewar  inferred  that  the  density  of  the  occluded 
hydrogen  may  be  as  great  as  0.623 ;  but  no  direct  measure  of 
the  density  has  ever  given  anything  approaching  this  figure, 
which  is  seven  times  the  maximum  observed  value  quoted 
in  the  table.  It  seems  fair  to  conclude,  therefore,  that  we 
are  to  explain  this  great  apparent  density  in  some  other  way. 
It  is  not  unlikely,  for  example,  that  the  occluded  hydrogen  is 
combined  with  the  palladium  so  as  to  form  a  hydride,  Pd2  H. 
Dewar  himself  suggests  this,  but  he  computes  the  density  of 
the  hydrogen  as  though  there  were  no  such  combination. 
So  far  as  I  am  aware,  there  is  no  sufficient  reason  for  suppos- 
ing that  the  volume  of  a  molecule  remains  unchanged  when 
this  molecule  combines  chemically  with  another  one;  and 
hence  the  high  density  obtained  by  Dewar  does  not  necessarily 
refute  the  computations  we  have  made.  Another  point  to 
which  I  should  like  to  call  your  attention  is,  that  the  numbers 
we  have  obtained  show  that  in  gases  under  ordinary  con- 
ditions the  average  distance  from  any  given  molecule  to  its 
nearest  neighbor  is  only  about  10  or  15  times  the  diameter  of 
a  molecule.  For  example,  let  us  conceive  the  molecules  of 
a  mass  of  carbon  dioxide  to  be  suddenly  arrested  in  their 
motions,  and  to  be  rearranged  so  as  to  be  spaced  at  uniform 
distances  from  one  another.  Then  let  this  idealized  gas  be 
reduced  in  volume  by  causing  the  molecules  that  compose 
it  to  approach  one  another  uniformly  until  they  touch.  The 
distance  from  the  center  of  one  molecule  to  the  center  of 
the  next  one  is  then  equal  to  the  diameter  of  a  molecule; 
and  the  volume  of  the  resulting  mass  is  .000843  of  the 
volume  of  the  original  gas.  Then  since  the  linear  dimensions 
of  similar  bodies  are  to  one  another  as  the  cube  roots  of 
the  corresponding  volumes,  we  have  the  proportion 


138  THE   MOLECULAR   THEORY   OF   MATTER. 

ORIGINAL    CENTRE   DISTANCE  :  DIAMETER    OF  A  MOLECULE 


=  Vl  :  V.000843 

from  which  it  follows  that  in  carbon  dioxide  at  0°  C.  and 
under  atmospheric  pressure,  the  average  distance  from  the 
center  of  any  one  molecule  to  the  center  of  its  nearest  neighbor 
is  about  10£  times  the  diameter  of  a  molecule.  Similar 
computations  for  the  other  gases  show  that  the  corresponding 
molecular  distance  is  about  14  diameters  in  hydrogen,  10 
diameters  in  nitrogen,  and  10^-  diameters  in  oxygen.  (These 
results  justify  the  remark  on  page  14,  where  the  popular 
comparison  of  molecular  dimensions  with  stellar  dimensions 
is  pronounced  erroneous.) 

Molecular  Diameters  by  Clausius's  Equation.  —  As  long 
ago  as  1858  Clausius  published  the  following  remarkable 
theorem  concerning  the  diameters  of  gas-molecules  :  If  the 
average  free  path  be  multiplied  by  8,  the  product  will  bear 
the  same  ratio  to  the  diameter  of  the  molecule  that  the  total 
space  containing  the  gas  bears  to  the  space  actually  occupied 
by  the  molecules.  At  the  time  this  theorem  was  published, 
Maxwell's  law  of  the  distribution  of  velocities  in  gases  had 
not  been  discovered,  and  Clausius,  in  his  investigations,  had 
assumed  the  molecules  to  have  substantially  the  same  velocities 
throughout  the  gas.  Maxwell's  subsequent  researches  made 
it  possible  to  take  account  of  the  differences  in  velocity  that 
exist  among  the  molecules  of  a  given  mass  of  gas,  and  when 
this  had  been  done  it  was  found  that  the  same  theorem  holds 
true,  except  that  in  the  place  of  the  numerical  factor  8  we 
must  use  6  V2  (=  8.485)  ;  so  that  in  its  corrected  form 
Clausius's  theorem  is  as  follows  :  * 


*  This  equation  is  readily  obtained  from  Clausius's  Kinetische  Theorie 
der  Gase  (1889-91),_page  65,  equation  (20),  by  multiplying  both  sides  of 

6  \l"2 
his  equation  by  -  »  and  observing  that  ^  TT  o-3  N  is  the  actual  volume  of 

<T 

the  molecules  that  exist  in  the  volume  V  of  the  gas. 


LORD  KELVIN'S  ELECTRICAL  METHOD. 


139 


where  \  is  the  average  free  path,  8  is  the  mean  diameter  of 
the  molecule,  and  e  is  the  actual  united  volume  of  the  molecules, 
that  exist  in  a  unit  volume  of  the  gas.  All  of  these  quantities,, 
except  8,  can  be  determined  by  methods  that  I  have  given  you 
this  evening ;  and  hence  equation  (52)  enables  us  to  calculate  8, 
the  diameter  of  the  molecule.  If  we  preserve  only  one  decimal 
place  in  the  numerical  factor  we  shall  have  8.5  instead  of 
8.485,  and  as  this  is  quite  a  sufficient  approximation  for  our 
present  purpose,  we  may  write  (52)  thus : 

8  =  8.5Ae.  (53) 

In  the  accompanying  table  I  have  collected  the  values  we  have 
obtained  for  A.  and  c,  and  the  values  of  8  that  are  obtained  by 
substituting  these  quantities  in  (53). 

MOLECULAR  DIAMETERS  BY  CLAUSIUS'S  METHOD. 


GAS. 

X 

e 

5 

Hydrogen  .    .  " 

.000  01690  cm. 

.00033 

47  X  10~9  cm. 

19X10-9  in. 

Nitrogen    .    . 

.000  00903 

.00087 

67X10-9 

27  X  10-9 

Oxygen  .    .    . 

.000  00964 

.00074 

61  X  10-9 

24X10~9 

Carbon  dioxide 

.000  00631 

.00072 

39X10-3 

15X10-9 

If  you  are  not  familiar  with  the  notation  used  in  the  last  two 
columns  I  will  explain  that  15  X  10~9,  for  example,  signifies 
that  15  is  to  be  divided  by  the  ninth  power  of  10.  Hence  the 
molecule  of  carbon  dioxide  has  a  diameter  of  .000,000,015  of 
an  inch,  and  if  67  millions  of  them  were  placed  in  contact 
with  one  another  in  a  straight  line,  the  row  thus  formed  would 
be  one  inch  long. 

Lord  Kelvin's  Electrical  Method.  —  Clausius's  method, 
which  I  have  just  explained  to  you,  purports  to  give  us  the 
actual  mean  diameter  of  a  molecule,  while  most  of  the  other 
methods  give  us  merely  minimum  or  maximum  limits  to  their 
sizes,  between  which  limits  their  true  size  must  lie.  Lord 


140      THE  MOLECULAR  THEORY  OF  MATTER. 

Kelvin's  electrical-potential  method  is  one  of  the  most  impor- 
tant of  those  that  have  been  proposed  to  fix  the  smallest 
admissible  size  of  molecules,  and  I  will  explain  it  to  you  as 
briefly  as  I  can,  following  his  own  words  as  closely  as  may  be.* 
He  considers  the  attraction  that  exists  between  plates  of 
copper  and  zinc,  when  they  are  put  in  metallic  connection  with 
each  other.  It  is  well  known  that  under  such  circumstances 
the  two  metals  assume  different  electrical  potentials,  and  by 
observing  their  difference  of  potential  we  can  compute  the 
attractive  force  that  they  exert  upon  each  other,  even  when 
this  force  is  so  small  that  it  cannot  be  weighed  or  otherwise 
determined  directly.  "  If  I  take  these  two  pieces  of  zinc  and 
copper  and  touch  them  together  at  the  two  corners,"  he  says, 
"  they  become  electrified,  and  attract  each  other  with  a  per- 
fectly definite  force,  of  which  the  magnitude  is  ascertained 
from  absolute  measurements  in  connection  with  the  well- 
established  doctrine  of  contact  electricity.  I  do  not  feel  it, 
because  the  force  is  very  small,  but  you  may  do  the  thing  in 
a  measured  way ;  you  may  place  a  little  metallic  knob  or  pro- 
jection of  T^(jVi»(Jtn  °^  a  centimeter  on  one  of  them,  and  lean 
the  other  against  it.  Let  there  be  three  such  little  metal  feet 
put  on  the  copper ;  now  touch  the  zinc  plate  with  one  of  them, 
and  turn  it  gradually  down  till  it  comes  to  touch  the  other 
two.  In  this  position,  with  an  air-space  of  TTJoVtJo^  °^  a 
centimeter  between  them,  there  will  be  positive  and  negative 
electricity  on  the  zinc  and  copper  surfaces  respectively,  of 
such  quantities  as  to  cause  a  mutual  attraction  amounting  to 
2  grammes  weight  per  square  centimeter.  The  amount  of 
work  done  by  the  electric  attraction  upon  the  plates  while 
they  are  being  allowed  to  approach  one  another  with  metallic 
connection  between  them  at  the  corner  first  touched,  till  they 
come  to  the  distance  of  TWooutn  °f  a  centimeter,  is  y^^Wo tns 
of  a  centimeter-gramme,  supposing  the  area  of  each  plate  to 
be  one  square  centimeter  . . .  Now  let  a  second  plate  of  zinc 
be  brought  by  a  similar  process  to  the  other  side  of  the  plate 
*  See  his  Popular  Lectures  and  Addresses,  Vol.  I,  page  168. 


LOED  KELVIN'S  ELECTRICAL  METHOD.  141 

of  copper  ;  a  second  plate  of  copper  to  the  remote  side  of  this 
second  plate  of  zinc,  and  so  on  till  a  pile  is  formed  consisting 
of  50,001  plates  of  zinc  and  50,000  plates  of  copper,  separated 
by  100,000  spaces,  each  plate  and  each  space  being  TTnjV<y  <y  th 
of  a  centimeter  thick.  The  whole  work  done  by  electric 
attraction  in  the  formation  of  this  pile  is  two  centimeter- 
grammes.  The  whole  mass  of  metal  is  eight  grammes.  Hence 
the  amount  of  work  is  a  quarter  of  a  centimeter-gramme  per 
gramme  of  metal.'7  By  taking  account  of  the  mechanical 
equivalent  of  heat,  and  of  the  specific  heats  of  zinc  and 
copper,  it  appears  that  the  amount  of  work  thus  done  by  the 


FIGS.  48  and  49.  —  ILLUSTRATING  LORD  KELVIN'S  ELECTRICAL  METHOD. 

electric  attraction  would  suffice  to  warm  the  pile  by  only 
Tsrsiyth  °f  a  Centigrade  degree.  "But  now  let  the  thickness 
of  each  piece  of  metal  and  of  each  intervening  space  be 
T (i o  o £o  o o  <j tn  of  a  centimeter,  instead  of  T<y ^ <y <y th-  Tne  work 
would  be  increased  a  millionfold  unless  TmF(yitr<T^(j^n  °^  a 
centimeter  approaches  the  smallness  of  a  molecule.  The  heat 
equivalent  would  therefore  be  enough  to  raise  the  temperature 
of  the  material  by  62°  C.  This  is  barely,  if  at  all,  admissible, 
according  to  our  present  knowledge,  or,  rather,  want  of  knowl- 
edge, regarding  the  heat  of  combination  of  zinc  and  copper. 
But  suppose  the  metal  plates  and  intervening  spaces  to  be 
made  yet  four  times  thinner,  that  is  to  say,  the  thickness  of 
each  to  be  ^oTn^inmTytf1  °^  a  centimeter.  The  work  and  its. 
heat  equivalent  will  be  increased  sixteenfold.  It  would. 


142      THE  MOLECULAR  THEORY  OF  MATTER. 

therefore  be  990  times  as  much  as  that  required  to  warm  the 
mass  by  one  degree  Centigrade,  which  is  very  much  more  than 
can  possibly  be  produced  by  zinc  and  copper  in  entering  into 
molecular  combination.  Were  there  in  reality  anything  like 
so  much  heat  of  combination  as  this,  a  mixture  of  zinc  and 
copper  powders  would  if  melted  in  any  one  spot,  run  together, 
generating  more  than  enough  heat  to  melt  them  both  through- 
out ;  just  as  a  large  quantity  of  gunpowder  if  ignited  in  any- 
one spot  burns  throughout  without  fresh  application  of  heat." 
This  argument  shows  that  it  would  not  be  possible  to  make 
plates  of  such  an  exceeding  thinness  without  splitting  mole- 
cules. If,  as  Lord  Kelvin  suggests,  a  temperature  elevation 
of  62°  C.  "is  barely,  if  at  all,  admissible,"  it  follows  that 
TtfTFfftaffnffth  °f  a  centimeter  is  the  minimum  admissible  size 
of  the  molecules  of  copper  and  zinc ;  or  else  (which  may  in 
reality  be  substantially  the  same  thing)  that  when  these 
metals  combine  in  an  alloy,  their  molecules  do  not  approach 
one  another  nearer  than  this.  Expressed  in  the  notation  of 
the  table  I  gave  you  a  few  moments  ago,  this  limit  is  10  X  10~9 
cm.,  or  4  X  10~9  in.  It  is  interesting  to  observe  the  substan- 
tial agreement  between  the  results  of  such  radically  different 
methods  as  Clausius's  and  Lord  Kelvin's,  especially  as  there 
is  no  a  priori  reason,  so  far  as  I  know,  for  supposing  that  the 
molecules  of  zinc  and  copper  are  anything  like  those  of  hydro- 
gen, oxygen,  nitrogen,  and  carbon  dioxide,  in  size  or  in  any 
•other  respect. 

Method  by  Camphor  Movements.  —  In  speaking  of  the 
molecular  theory  of  liquids  I  told  you  something  about  the 
motion  of  particles  of  camphor  on  the  surface  of  water,  and  I 
<called  your  attention  to  the  fact  that  to  ensure  success  in  the 
experiment  it  is  necessary  to  have  the  water-surface  perfectly 
•clean.  The  least  trace  of  oily  matter  checks  the  camphor- 
:movements  quite  sensibly ;  and  this  fact  has  suggested  a 
simple  method  of  determining  a  maximum  limit  to  the  size  of 
molecules.  Instead  of  explaining  this  method  in  the  abstract, 


THE    SURFACE    TENSION    METHOD.  143 

I  will  tell  you  how  Lord  Kayleigh  put  it  into  practice.*  He 
provided  a  circular  "  sponge  bath,"  84  centimeters  in  diam- 
eter, and  having  cleaned  it  carefully  he  allowed  pure  water  to 
flow  into  it  to  a  depth  of  several  inches.  Camphor  scrapings 
were  deposited  on  the  water  at  several  places  widely  removed 
from  one  another,  and  these  at  once  exhibited  vigorous  move- 
ments. A  fine  platinum  wire  was  then  weighed,  touched  with 
olive  oil,  and  re-weighed ;  after  which  it  was  cautiously  applied 
to  the  water-surface.  The  oil  spread  over  the  water,  and  the 
experimenter  noted  the  effect  on  the  camphor  particles.  In 
one  case  in  which  he  notes  that  there  was  "just  about 
enough"  oil  to  stop  the  movements,  the  weighings  of  the 
platinum  wire  indicated  that  0.81  of  a  milligram  of  oil  had 
been  added  —  that  is,  0.00081  of  a  gramme.  Now  a  cubic 
centimeter  of  olive  oil  weighs  about  0.9  of  a  gramme ;  and 
hence  it  follows  that  the  volume  of  oil  applied  was  0.00081 
-f-  0.9  =  0.00097cc.  The  area  of  the  sponge  bath  being  .7854 
X  S42  square  centimeters,  the  volume  of  the  film  of  oil  that 
sufficed  to  still  the  movements  was  .7854  X  842  X  t,  where  t  is 
the  unknown  thickness  of  the  film.  Assuming  that  the 
volume  of  olive  oil  remains  constant  under  the  given  con- 
ditions, we  have 

.7854  X  842  X  t  =  0.0009  cc., 

whence  t  =  163  X  10~9  centimeters,  or  65  X  10~9  inches.  Now 
we  do  not  know  that  this  film  of  oil  is  just  one  molecule  thick, 
but  we  do  know  that  it  is  not  thinner  than  that.  Hence  Lord 
Kayleigh's  method  gives  us  a  maximum  limit  to  the  size  of  a 
molecule  of  olive  oil. 

The  Surface  Tension  Method.  — This  method  is  due,  I 
believe,  to  Lord  Kelvin ;  but  I  arn  going  to  give  you  a  pre- 
sentation of  it  that  is  slightly  different  from  his.  Let  us 
consider  a  mass  of  liquid  whose  surface  tension  is  S  grammes 

*  Lord  Rayleigh's  paper,  read  before  the  Royal  Society  in  March, 
1890,  is  reproduced  in  Lord  Kelvin's  Popular  Lectures  and  Addresses. 


144     THE  MOLECULAR  THEORY  OF  MATTER. 

per  linear  centimeter.  Now  if  the  surface  of  this  liquid  be 
extended  in  any  manner  until  it  increases  by  the  small  amount 
A^4,  the  work  done  against  the  surface  tension  will  be  S  .  &A. 
Lord  Kelvin  has  shown,  however,  that  a  liquid  is  cooled  by 
extending  its  surface,  and  hence  the  entire  quantity  of  energy 
required  to  increase  the  surface  is  not  fairly  represented  by 
S.  A^4,  for  we  are  assisted  by  the  sensible  heat  that  disappears 
during  the  process  of  extension.  The  results  of  this  thermal 
effect  have  been  investigated  by  thermodynamical  methods,* 
and  it  has  been  proved  that  if  we  keep  the  temperature 
constant  by  adding  heat  from  without,  and  thus  maintain  the 
heat-energy  of  the  liquid  constant,  the  entire  expenditure  of 
energy  required  to  increase  the  surface  by  an  amount  AJ.  can 
be  written  in  the  form  mS.  A  A,  where  m  is  a  certain  numerical 
factor  dependent  upon  the  nature  of  the  liquid,  and  upon  the 
constant  temperature  at  which  the  experiment  is  performed. 
For  water  at  ordinary  temperatures  m  =  1.5,  approximately. 
What  we  have  really  done,  in  extending  the  surface  of  the 
liquid,  is  to  bring  more  particles  of  it  out  from  the  interior  : 
we  have,  in  fact,  brought  to  the  surface  a  sufficient  number  of 
molecules  to  form  a  layer  whose  area  is  A-4,  and  whose  thick- 
ness is  the  diameter  of  a  molecule  (which  we  will  represent 
by  8).  The  volume  of  this  layer  is  8  .  A^4,  and  its  weight  is 
w  .  8  .  A^4,  where  w  is  the  weight  of  a  unit  volume  of  the  liquid. 
Now  we  have  already  found  the  work  required  to  bring  a  unit 
weight  of  liquid  to  the  surface,  and  I  have  given  you  a  table 
of  values  of  it,  for  four  different  liquids  [page  101].  The 
work  that  would  have  to  be  expended  in  order  to  bring  to  the 
surface  the  layer  we  are  considering  is  found  by  multiplying 

W 

w.8.  &A  by  the  value  of  —  for  the  corresponding  liquid  ;  and 

2i 

since  it  must  also  be  equal  to  mS  .  AJ,  we  have  the  equation 


*  The  investigation  is  given  in  the  Appendix. 


RANGE   OF   MOLECULAR   ATTRACTION. 


145 


or,  dividing  by  &A  and  solving  for  8, 

2mS 


(54) 


From  this  equation  I  have  prepared  the  accompanying  table, 
which  gives  the  values  of  8  for  those  liquids  for  which  we 
have  previously  computed  W.  The  chief  weakness  of  the 
surface-tension  method  consists  in  the  underlying  assumption 
that  the  surface  of  a  liquid  is  a  perfectly  definite  thing  — 
which,  as  I  have  endeavored  to  illustrate  in  Fig.  27,  is  far 


MOLECULAR  DIAMETERS  BY  THE  SURFACE  TENSION  METHOD. 


LIQUID. 

8 

m* 

w 

W 

8 

Water        '  ^  -  ' 

.074 

1.57 

1.00 

(cm.-gms.) 
18.6  XlO6 

(cm.) 
12  X  10-9 

Alcohol      

.026 

2.36 

0.82 

3.5X106 

43X10-9 

Bisulphide  of  Carbon    .     . 
Mercury                         . 

.033 
551 

2.16 
1.17 

1.29 
13.60 

1.8  XlO6 
2.  5  XlO6 

61  X  lO-9 
38X10-9 

from  being  the  case.  It  is  true  that  Lord  Kelvin  does  not 
push  his  method  to  the  extreme  to  which  I  have  carried  it, 
but  that  he  uses  it  only  to  obtain  a  general  estimate  of  the 
order  of  magnitude  of  a  molecule.  I  have  chosen  to  extend 
the  method,  in  order  that  its  limitations  may  appear  more 
clearly. 

Quincke's  Determination  of  the  Range  of  Molecular 
Attraction. — M.  Plateau  has  described  Quincke's  "wedge 
method "  very  clearly,  and  I  will  quote  the  description  of  it 
that  he  gives  :f  "When  a  liquid  lies  against  a  solid  vertical 
wall  that  it  does  not  wet,  it  makes  with  this  wall  an  angle 
whose  magnitude  depends,  as  is  well  known,  both  on  the 

*  For  the  values  of  m,  see  Appendix. 

t  Statique  Exp^rimentale  et  TMorique  des  Liquides,  etc.,  Vol.  I,  page 
211. 


146  THE   MOLECULAR   THEORY   OF   MATTER. 

nature  of  the  solid  and  on  that  of  the  liquid.  This  being 
granted,  let  us  conceive  that  upon  one  of  the  faces  of  a  glass 
plate  there  is  deposited  a  wedge-like  layer  (as  the  author  calls 
it)  of  some  other  solid  substance,  the  edge  of  the  wedge  being 
vertical,  and  its  thickness,  exceedingly  small  near  this  edge, 
increasing  uniformly  and  by  insensible  gradations  as  we  pass 
away  from  it.  If  a  liquid  is  now  allowed  to  rest  against  the 
plate  so  as  to  touch  the  glass  along  part  of  its  boundary  and 
the  wedge-shaped  layer  at  some  other  part,  the  angle  of  con- 
tact that  the  liquid  makes  with  the  plate  will  vary  as  we  pass 
away  from  the  edge  of  the  wedge  ;  for  in  the  neighborhood  of 
this  edge  the  mutual  molecular  action  between  the  glass  and 
the  liquid  will  still  make  itself  felt.  But  after  we  have 
reached  the  distance  at  which  the  thickness  of  the  wedge 
becomes  equal  to  the  radius  beyond  which  this  mutual  action 
is  insensible,  the  angle  of  contact  will  become  constant,  as  it 
will  no  longer  depend  upon  anything  but  the  mutual  action 
between  the  liquid  and  the  substance  of  which  the  wedge  is 
composed.  M.  Quincke  has  succeeded  in  obtaining,  upon 
glass,  wedge-shaped  layers  of  different  substances,  and  he  has 
determined,  in  each  case,  the  thickness  that  fulfills  the  con- 
dition indicated.  With  a  wedge  of  metallic  silver,  the  liquid 
being  water,  he  has  found  I  >.0000054  cm.,  I  being  the  thick- 
ness in  question;  with  a  wedge  of  sulphide  of  silver,  and 
mercury  as  the  liquid,  he  found  1= . 0000048  cm.;  with 
mercury  and  a  wedge  of  iodide  of  silver,  1  = . 0000059  cm.; 
and  with  mercury  and  a  wedge  of  collodion,  I  <  .0000080  cm. 
M.  Quincke  concludes  from  his  experiments  that  one  may 
adopt,  as  the  mean  value  of  the  radius  of  sensible  molecular 
attraction  in  these  cases,  l  =  . 0000050  cm." 

Other  Methods  of  Investigation.  —  Many  other  ingenious 
methods  have  been  devised  for  determining  the  sizes  of  mole- 
cules and  the  radius  of  sensible  molecular  attraction,  but  I 
can  consider  only  a  few  of  them.  First  of  all,  we  must  not 
•omit  to  mention  the  result  obtained  by  Plateau  himself.  He 


OTHEK    METHODS    OF   INVESTIGATION.  147 

found,  from  a  study  of  soap-bubbles,  that  the  limiting  thick- 
ness of  a  film  of  his  glycerine  liquid  —  that  is,  the  thickness 
.at  which  the  film  becomes  unstable  —  is  about  .0000114  cm.; 
and  as  he  believed  that  the  film  would  be  unstable  when  its 
thickness  was  reduced  to  twice  the  radius  of  sensible  molecular 
attraction,  he  concluded  that  the  value  of  this  radius,  in  the 
liquid  composing  his  bubbles,  was  .0000057  cm.  Maxwell, 
however,  has  endeavored  to  show  that  a  liquid  film  is  stable 
until  its  thickness  is  reduced,  not  to  double  the  radius  of 
attraction,  but  to  that  radius  itself.  If  his  theory  is  correct, 
we  must  conclude  from  Plateau's  data  that  the  radius  of 
sensible  attraction,  in  the  glycerine  liquid,  is  .0000114  cm. 
More  recently  the  limiting  thickness  of  soap-films  has  been 
accurately  measured  by  Eeinold  and  Eticker,  who  find  it  to 
be  about  .0000012  cm.  Oberbeck  found  that  a  coating  of 
metal  .0000003  cm.  thick  is  sufficient  to  polarize  platinum 
—  that  is,  this  thickness  is  sufficient  to  alter  the  character  of 
the  surface  of  platinum  so  that  a  distinct  difference  in 
electrical  potential  is  observed  between  a  plate  so  prepared 
and  another  plate  of  the  pure  metal.  Lord  Kelvin  found 
that  "  a  quite  infinitesimal  whiff  of  iodine  vapor  "  is  sufficient 
to  alter  the  surface  of  a  silver  plate  so  that  a  difference  in 
potential  is  observable ;  but  I  do  not  know  that  he  has  given 
any  numerical  estimate  of  the  thickness  of  the  resulting  film 
of  iodide  of  silver.  Wiener  found  that  a  film  of  silver 
.000.000,02  cm.  thick  produces  a  sensible  effect  on  the  phase 
of  reflected  light.  He  also  found  that  when  a  silver  plate  is 
reduced  in  thickness  to  .0000012  cm.,  it  no  longer  produces 
the  same  effect  on  the  phase  of  reflected  light  that  a  thick 
plate  does.  There  is  no  very  close  agreement  among  these 
various  results,  nor  could  we  reasonably  expect  that  there 
would  be,  when  the  quantities  measured  are  so  imperfectly 
defined,  and  the  methods  and  substances  used  are  so  different. 
Nevertheless,  the  results  are  all  of  the  same  general  order  of 
magnitude,  and  when  taken  in  connection  with  those  obtained 
by  the  methods  that  I  have  already  described,  they  suffice  to 


148 


THE  MOLECULAR  THEOHY  OF  MATTER. 


give  us  a  very  good  general  idea  of  the  sizes  of  molecules, 
and  of  the  distance  at  which  molecular  forces  cease  to  be 
sensible. 

Number  of  Molecules  in  a  Unit  Volume  of  Gas.  —  Accord- 
ing to  Avogadro's  law,  all  gases  contain  the  same  number  of 
molecules  per  unit  volume,  when  they  are  subject  to  the  same 
conditions  ;  and  it  will  be  interesting  to  calculate  this  number 
for  some  given  temperature  and  pressure.  If  we  continue  to 
regard  the  molecules  of  a  gas  as  spherical,  the  volume  of  a 
single  molecule  is  £  7r88 ;  and  if  -ZV  is  the  number  of  them  in  a 
cubic  centimeter  of  the  gas,  the  combined  volume  of  all  the 
molecules  in  this  cubic  centimeter  will  be  £  NvS8.  But  this  is 
what  we  have  called  the  "  aggregate  molecular  volume,"  and 
have  denoted  by  e.  Hence 


or      = 


(55) 


By  substituting  in  this  equation  the  values  of  e  and  8  that  we 
have  found  for  various  gases,  we  obtain  the  values  of  N  given 
in  the  table. 


NUMBER  OF  MOLECULES  IN  A  UNIT  VOLUME  OF  GAS,  AT  0°C.,  AND 
ATMOSPHERIC  PRESSURE. 


GAS. 

NUMBER  OF  MOLECULES. 

In  a  Cubic  Centimeter. 

In  a  Cubic  Inch. 

Hydrogen    
Nitrogen      

6.1X1018 
5.5X1018 
6.2X10" 
23.2X1018 

95X1018 
86X1018 
97  X  1018 
363  x  10i8 

Oxygen  
Carbon  dioxide                    .     . 

These  numbers  are  as  nearly  equal  as  could  well  be  expected. 
It  must  be  remembered  that  in  the  determination  of  N  we 
have  the  accumulated  effects  of  all  the  errors  that  have  been 


ILLUSTRATIONS    OF   MOLECULAR    MAGNITUDES.     149 

committed  in  obtaining  the  coefficients  of  viscosity,  the  free 
paths,  the  "  aggregate  molecular  volumes/'  and  the  molecular 
diameters.  The  values  of  ^V  for  hydrogen,  nitrogen,  and 
oxygen  agree  among  themselves  remarkably,  but  the  value  for 
carbon  dioxide  is  apparently  too  large.  You  will  observe 
that  in  determining  e  I  have  used  Mr.  Sutherland's  data  for 
the  first  three  gases,  and  Clausius's  data  for  the  last  one.  If 
we  take  Mr.  Sutherland's  results  for  all  the  gases,  we  should 
have,  for  carbon  dioxide,  €  =  .00117,*  which  is  about  sixty 
per  cent  greater  than  the  value  given  by  Clausius's  equation 
(30).  By  using  Mr.  Sutherland's  value  of  e  in  equation  (53) 
we  find  8  =  63  X  10~9  cm. ;  and  a  further  substitution,  in  (55), 
gives  N=  8.9  X  1018*  as  the  number  of  molecules  in  a  cubic 
centimeter  of  the  gas,  under  the  stated  conditions.  This 
agrees  much  better  with  the  results  given  in  the  table  for 
hydrogen,  nitrogen,  and  oxygen ;  and,  as  these  four  values  of 
JVhave  been  deduced  from  independent  observations,  I  think 
we  may  reasonably  conclude  that  Clausius's  theorem  will  give 
us  a  fairly  accurate  knowledge  of  molecular  diameters,  and 
hence  also  of  N,  when  the  mathematical  investigation  of  the 
kinetic  theory  of  gases  has  been  carried  far  enough  to  enable 
us  to  construct  a  theoretically  perfect  gas  equation,  from 
which  to  deduce  accurate  values  of  e.  At  present  I  think  we 
can  only  say  that  at  0°  C.  and  under  one  atmosphere  pressure, 
one  cubic  centimeter  of  gas  contains  something  like  7,000,- 
000,000,000,000,000  molecules.  This  corresponds  to  about 
110,000,000,000,000,000,000  per  cubic  inch. 

Illustrations  of  Molecular  Magnitudes.  —  I  have  given  you 
a  lot  of  numbers  with  strings  of  ciphers  attached  to  them, 
sometimes  running  away  off  to  the  right,  and  at  other  times 
off  to  the  left.  I  should  like,  now,  to  give  you  a  few  homely 

*  See  the  Philosophical  Magazine  for  March,  1893,  page  232.  The 
quantity  that  I  have  denoted  by  e  is  found  by  dividing  Mr.  Sutherland's  ft 
by  the  volume  of  a  gramme  of  the  corresponding  gas,  measured  at  0°  C. 
and  one  atmosphere,  and  taking  f  of  the  quotient.  In  the  case  of  CO2 
we  have  c  =f  X  (0.692  -j-  505)  =.00117. 


150      THE  MOLECULAR  THEORY  OF  MATTER. 

illustrations  of  what  these  numbers  mean.  We  have  found 
that  in  gaseous  bodies  the  molecules  are  moving  with  speeds 
comparable  with  that  of  a  cannon  ball  —  indeed,  we  found 
that  the  average  velocity  of  a  hydrogen  molecule,  at  0°  C., 
materially  exceeds  a  mile  a  second,  which  speed  has  never 
been  attained  with  the  best  modern  artillery.  We  have  found 
that  the  average  distance  that  a  gas-molecule  travels,  under 
ordinary  circumstances,  between  successive  collisions  with  its 
fellows,  is  only  a  few  millionths  of  an  inch,  —  a  distance  far 
too  small  to  be  seen  with  certainty  even  by  the  most  powerful 
microscopes.  The  number  of  collisions  experienced  by  a 
single  molecule  of  gas  in  one  second  was  also  found  to  be  very 
great.  Every  nitrogen  molecule,  for  example,  is  struck  by  its 
fellows,  under  the  standard  conditions,  something  over  5,000,- 
000,000  times  per  second.  It  would  take  you  fifty-three 
years  to  count  that  many,  if  you  should  count  three  every 
second,  and  twenty-four  hours  every  day.  We  have  found 
that  molecules  are  amazingly  small  —  so  small,. for  example, 
that  if  every  man,  woman,  and  child  in  this  world  were  to  lay 
down  a  molecule  of  carbon  dioxide  so  that  all  these  molecules 
should  lie  in  a  straight  line,  and  each  should  touch  its  neigh- 
bors, the  row  thus  formed  would  hardly  be  more  than  a  yard 
long.*  A  cube  formed  of  a  hundred  thousand  million  mole- 
cules of  hydrogen  in  contact  with  one  another  would  hardly 
be  visible  in  the  finest  microscopes  that  we  have.  The 
number  of  molecules  in  a  cubic  inch  of  gas  at  the  freezing 
point  and  under  atmospheric  pressure  is  so  great,  that  not- 
withstanding the  fact  that  it  would  take  about  55,000,000 
hydrogen  molecules  to  make  a  row  one  inch  long,  and  that 
only  one  three-thousandth  of  the  volume  of  the  gas  is  really 
filled  by  matter  (the  rest  being  vacuous  space  around  the 
molecules),  if  all  the  molecules  that  exist  in  a  cubic  inch  of 
this  gas  were  placed  in  a  row,  touching  one  another,  they 
would  form  a  line  about  32,000,000  miles  long,  or  long  enough 

*  According  to  the  Statesman's  Year-Book^  the  population  of  the  world, 
in  1890,  was  about  1,468,000,000. 


THE   CONSTITUTION   OF   MOLECULES.  151 

to  wind  around  the  earth  more  than  a  thousand  times.  If  the 
molecules  contained  in  a  cube  whose  edge  is  one  inch  and  a 
half  were  similarly  arranged,  they  would  reach  from  the  earth 
to  the  sun.  To  state  the  same  thing  in  still  another  way,  let 
me  say  that  if  the  molecules  in  a  cubic  inch  of  gas  (under  the 
assumed  conditions  of  temperature  and  pressure)  were  spread 
out  uniformly  so  as  to  form  a  single  sheet  or  layer,  and  if 
they  were  distributed  so  that  their  average  distance  from 
center  to  center  was  about  the  same  as  the  corresponding 
distance  between  the  letters  on  this  page,  then  the  layer  in 
question  would  cover  all  the  six  continents  of  the  earth,  six 
times  over.  It  would  be  easy  to  investigate  the  speed  of 
rotation  of  molecules  in  gases,  by  the  aid  of  Boltzmann's 
theorem,  and  to  show,  by  making  certain  assumptions  about 
their  shapes,  that  hydrogen  molecules  (for  example)  rotate  so 
swiftly  that  they  perform  one  entire  revolution  while  light, 
with  its  prodigious  velocity  of  186,000  miles  per  second,  is 
traveling  a  few  millionths  of  an  inch.  It  might  also  be 
interesting  to  speculate  on  the  tensile  strength  of  a  substance 
that  could  hold  together  while  rotating  with  so  fearful  a 
speed.  I  have  not  discussed  these  points,  however,  for  I  was 
afraid  that  if  I  should  do  so,  I  could  not  help  conveying  the 
impression  that  we  know  far  more  about  them  than  we  do. 
Some  day  it  may  be  profitable  to  theorize  about  these  things, 
but  certainly  the  time  is  not  yet  come. 

VI.     THE   CONSTITUTION  OF  MOLECULES. 

Preliminary  Remarks. — We  are  now  about  to  leave  the 
field  of  knowledge  altogether,  and  to  enter  the  vast  domain  of 
speculation ;  for  I  think  I  can  safely  say  that  at  present  we 
do  not  possess  the  least  particle  of  positive  information  con- 
cerning the  constitution  of  molecules.  I  have  even  had 
doubts  about  the  propriety  of  discussing  the  subject  at  all,  so 
profound  is  our  ignorance  of  it ;  but  upon  more  mature 
reflection  it  seemed  hardly  proper  to  omit  all  mention  of  the 


152      THE  MOLECULAR  THEORY  OF  MATTER. 

views  that  have  been  advanced  by  physicists  from  time  to 
time,  especially  as  we  have  made  certain  tacit  assumptions 
about  the  constitution  of  molecules  in  the  preceding  sections 
—  assumptions  whose  uncertainty  should  at  least  be  pointed 
out.  The  chief  difficulty  before  the  philosopher  who  would 
investigate  the  structure  of  molecules  lies  in  the  fact  that  it 
is  impossible  to  see  or  measure  a  molecule  directly.  We  know 
them,  not  individually,  but  only  in  the  aggregate  ;  and  we 
can  infer  their  structure  only  by  observing  the  gross  results 
of  their  interaction  in  large  numbers.  Our  situation  may  be 
best  illustrated,  perhaps,  by  imagining  a  huge  being  who 
could  observe  only  the  general  trend  of  events  upon  the  earth 
—  the  march  of  civilization  and  the  rise  and  fall  of  empires  — 
and  who  desired  to  infer  from  these  data  the  anatomy  and 
mental  characteristics  of  the  invisible  creatures  whose  indi- 
vidual acts,  when  summed  up,  had  given  rise  to  these 
phenomena.  It  is  obvious  that  in  either  case  the  problem  is 
of  enormous  difficulty,  and  that,  to  a  certain  extent,  it  may 
prove  to  be  indeterminate.  "VVe  must  proceed  by  making 
various  assumptions  about  the  constitution  of  molecules,  and 
we  must  then  rigorously  follow  out  the  results  of  these 
assumptions  and  compare  them  with  the  facts.  Those  that 
are  found  to  disagree  with  the  facts  will  then  be  rejected,  and 
the  others  will  be  tentatively  retained  until  new  and  crucial 
facts  may  be  discovered,  which  shall  make  a  further  rejection 
possible.  By  such  a  process  of  exclusion  we  may  hope  to 
arrive,  some  day,  at  a  fair  knowledge  of  the  structure  of 
molecules ;  but  the  progress  of  the  molecular  theory  in  this 
direction  is  likely  to  be  extremely  slow,  on  account  of  the 
mathematical  difficulties  that  continually  present  themselves, 
and  which  can  be  overcome  only  by  a  great  amount -of  patient 
labor. 

General  Facts  to  be  Explained.  —  The  successful  molecular 
theory  —  or  "  final  theory,"  as  we  may  call  it,  —  must  explain 
a  vast  range  of  phenomena,  extending  all  the  way  from  those 


GENERAL  FACTS  TO  BE  EXPLAINED.       153 

that  thrust  themselves  upon  us  in  our  everyday  life  to  those 
more  recondite  ones  that  can  be  observed  only  in  the  labora- 
tory, under  specially  favorable  conditions.  Some  physicists 
demand  that  the  final  theory  shall  explain  even  the  inertia  of 
matter ;  but  it  seems  to  me  that  this  is  asking  too  much,  and 
that  such  a  demand  implies  an  imperfect  understanding  of 
what  "  explanation  "  means.  To  "  explain  "  a  fact  is  to  show 
that  it  is  a  necessary  consequence  of  other  facts  that  are  more 
readily  grasped  by  the  human  understanding ;  and  as  inertia 
is  perhaps  the  most  fundamental  of  the  known  properties  of 
matter,  it  seems  as  though  any  attempted  explanation  of  it 
would  only  involve  us  in  a  sea  of  words  and  ink,  without 
enlightening  us  in  the  least.  Elasticity,  however,  is  certainly 
capable  of  explanation.  When  it  is  manifested  by  bodies  of 
sensible  size,  we  refer  it  to  the  interaction  of  the  molecules  of 
which  the  bodies  are  composed ;  but  when  it  is  manifested  by 
the  molecules  themselves,  some  further  explanation  is  required. 
I  have  told  you  that  molecules  are  perfectly  elastic ;  but  I 
have  merely  postulated  the  elasticity,  and  have  not  attempted 
to  explain  it.  We  shall  see,  presently,  that  Lord  Kelvin's 
vortex  theory  affords  a  mechanical  explanation  of  the  elasticity 
of  molecules,  and  although  I  should  not  like  to  say  that  his 
explanation  is  the  correct  one,  it  proves  to  us,  at  least,  that 
.such  an  explanation  is  possible.  The  final  molecular  theory 
must  unravel  the  mysteries  of  chemical  and  physical  attraction, 
and  must  also  show  why  an  atom  of  A  has  a  powerful  affinity 
for  an  atom  of  B,  but  is  comparatively  indifferent  towards  an 
atom  of  C.  It  must  unify  the  conceptions  of  chemistry  and 
physics,  and  consolidate  these  sciences  into  one  grand  Science 
of  Matter.  It  will  also  have  to  explain  the  possibility  of 
such  enormous  and  comparatively  stable  aggregates  of  atoms 
as  occur  in  proteid  bodies  and  gums,  whose  molecular  weights, 
in  some  instances,  are  believed  to  be  as  high  as  13,000  or 
14,000.  These  are  formidable  things  to  require  of  an  infant 
theory,  and  yet  no  theory  which  does  not  cover  them  all  can 
possibly  hope  for  final  acceptance.  It  may  be  well  to  indicate 


154  THE    MOLECULAR   THEORY   OF   MATTER. 

some  of  the  general  facts  of  chemistry  and  physics  that  are 
likely  to  serve  as  stepping-stones  towards  the  great  theory  of 
matter  that  future  generations  will  undoubtedly  evolve,  and  I 
shall  therefore  refer  briefly  to  Dulong  and  Petit's  law,  to 
Prout's  hypothesis,  to  the  periodic  law  of  Mendeleieff  and 
Meyer,  and  to  the  theory  of  radiation. 

Dulong  and  Petit's  Law.  —  In  the  early  part  of  the  present 
century  two  distinguished  French  physicists,  MM.  Dulong 
and  Petit,  announced  that  the  specific  heats  of  certain 
elements  upon  which  they  had  experimented  are  inversely 
proportional  to  the  respective  atomic  weights  of  those 
elements,  and  from  this  they  concluded  that -the  amount  of 
heat  required  to  raise  the  temperature  of  N  atoms  by  1°  is 
the  same  for  all  the  elementary  bodies.  This  remarkable 
generalization  did  not  meet  with  universal  and  immediate 
acceptance,  because  it  failed  in  numerous  cases*  unless  the 
atomic  weights  of  the  corresponding  elements  were  changed 
somewhat  from  the  values  that  had  been  assigned  to  them 
from  purely  chemical  considerations.  Moreover,  it  could  not 
possibly  be  an  exact  law,  because  the  specific  heats  of  bodies 
are  not  constant,  but  vary  with  the  temperature,  and  some- 
times to  a  considerable  extent.  Subsequent  experimenters 
have  paid  great  attention  to  Dulong  and  Petit's  law,  however, 
and  now  that  the  atomic  weights  of  the  more  familiar 
elements  have  been  pretty  well  determined,  the  law  is  found 
to  be  surprisingly  near  the  truth.  Dr.  L.  Meyer  gives  a  list 
of  fifty  elements  which  accord  with  it  very  well,  and  I  have 
selected  ten  of  them,  almost  at  random,  to  show  you  its  wide 
applicability,  and  the  order  of  its  accuracy.*  The  atomic 
weights  in  these  elements  range  from  7  to  240,  and  yet  when 
we  multiply  each  of  them  by  the  corresponding  specific  heat 
we  find  that  the  product  remains  constant,  or  nearly  so. 
Furthermore,  the  small  variations  that  do  occur  do  not  appear 

*  Dr.  Lothar  Meyer,  Modern  Theories  of  Chemistry  (London,  Long- 
mans, Green  &  Company,  1888),  page  73. 


DULONG    AND    PETIT  S    LAW. 


155 


to  follow  any  particular  law,  but  have  rather  the  character  of 
"errors  of  observation";  and  although  they  are  quite  too 
large  to  be  attributed  to  this  cause,  their  irregularity  forces 

ILLUSTRATIONS  OF  DULONG  AND  PETIT' s  LAW. 


S  S 

0 

p 

1  w 

£H 

e 

ELEMENT. 

Is 

1  1 

ELEMENT. 

H    S 

|  | 

Q 

^  &• 

" 

PH 

<  £ 

CO 

PH 

Lithium  . 

7.0 

.941 

6.6 

Antimony  .     . 

120 

.0508 

6.1 

Aluminium  .     . 

27.0 

.214 

5.8 

Tungsten  .     . 

184 

.0334 

6.1 

Potassium    .     . 

39.0 

.166 

6.5 

Gold      .     .     . 

196 

.0324 

6.4 

Copper    .     .     . 

63.2 

.0952 

6.0 

Bismuth     .     . 

207 

.0308 

6.4 

Silver      .     .     . 

107.7 

.0570 

6.1 

Uranium    .     . 

240 

.0277 

6.6 

upon  us  the  conviction  that  Dulong  and  Petit's  law  is  not  a- 
mere  "  first  approximation  "  to  the  relation  between  specific 
heat  and  atomic  weight,  but  that  it  expresses  that  relation 
accurately,  and  that  the  outstanding  differences  are  due  to 
causes  which  are  not  functions  of  the  atomic  weights  alone. 
Dulong  and  Petit's  law  has  been  thought,  by  some,  to  indicate 
that  matter,  in  the  last  analysis,  is  of  only  one  kind.  To  my 
own  mind  the  law  seems  to  controvert  this  hypothesis,  rather 
than  to  sustain  it ;  for  if  there  is  really  but  one  ultimate  kind 
of  matter,  we  might  indeed  expect  to  find  the  specific  heats  of 
bodies  equal,  but  I  think  we  should  hardly  look  for  them  to 
be  inversely  proportional  to  the  molecular  weight.  Dulong 
and  Petit's  law  seems  rather  to  be  an  experimental  indication 
that  some  modification  of  Boltzmann's  law  of  the  partition  of 
kinetic  energy  in  the  molecules  of  gases  will  also  be  found  to 
be  true  of  the  molecules  of  liquids  and  solids.  To  illustrate, 
let  us  consider  a  gas  in  which  the  effects  of  intermolecular 
attraction  are  insensible.  Maxwell  has  shown  that  if  two* 
such  gases  have  the  same  temperature,  their  kinetic  energies.; 
of  translation  (per  molecule)  will  also  be  equal  [page  35]i. 
Since  this  is  true  for  any  temperature,  it  follows  that  if  the 
two  gases  be  raised  through  the  same  range  of  temperature 


156      THE  MOLECULAR  THEORY  OF  MATTER. 

(say  from  ti  to  £2),  the  increase  in  the  average  kinetic  energy 
of  translation  will  be  the  same  in  each  case.  For  the  sake  of 
clearness  let  us  name  the  two  gases  "A"  and  "JB,"  respect- 
ively ;  and  let  us  denote  by  k  the  average  increase  in  kinetic 
energy  of  translation  experienced  by  a  molecule  of  either  gas 
when  the  temperature  is  raised  from  t°  to  (£-|-l)0.  Kow  if 
n-L  and  n2  are  the  number  of  degrees  of  freedom  of  the  mole- 
cules of  A  and  B,  respectively,  then  the  total  increase  in 
kinetic  energy  of  N  molecules,  when  the  temperature  is  raised 
from  t°  to  (<  +  l)°,  will  be 

^  •  Nfc  and  ^  •  Nk, 
o  o 

respectively.  But  since  we  assume  that  the  change  in 
potential  energy  is  insensible,  it  follows  that  these  quantities 
are  equal  to  s^W^  and  s2W2,  respectively,  where  s  is  the 
specific  heat  of  the  gas,  and  W  is  the  weight  of  N  molecules. 
Hence, 

^nlNk  =  s1Wl)  and  %n2Nk  =  s2W2.  (56) 

Now  although  we  do  not  know  N  with  any  considerable 
approach  to  accuracy,  we  do  know  that 

and   W 


where  iv±  and  w2  are  the  weights  of  a  molecule  of  A  and  B, 
respectively.  Substituting  these  values  of  TFi  and  W2  in 
(56)  and  dividing  by  N,  we  have 

^nik  =  slWi  and  ^  n2k  =  s2w2.     . 
Hence,  finally, 

S1Wl_S2W2 

-  \  (ot  ) 

W!         n2 

and  this,  I  imagine,  is  the  true  form  of  Dulong  "and  Petit's 
law.  As  the  absolute  values  of  w  are  not  known  with  pre- 
cision, we  may  advantageously  use,  instead  of  them,  those 
relative  values  which  chemists  call  the  "molecular  weights." 
^Equation  (57)  may  be  written 

sw  =  Cn}  (58) 


DULONG   AND   PETIT  S   LAW. 


157 


where  s  represents  specific  heat,  w  represents  molecular 
weight,  n  is  the  number  of  degrees  of  freedom  of  a  molecule, 
and  C  is  a  constant.  You  will  notice  that  this  differs  from 
Dulong  and  Petit's  law,  as  ordinarily  stated,  in  the  intro- 
duction of  n  in  the  second  member.  It  may  be  interesting  to 
determine  C  by  examining  some  of  those  gases  and  vapors 
for  which  n  has  been  determined  by  means  of  equation  (22). 


DETERMINATION  OF  C. 


SUBSTANCE. 

SPECIFIC 
HEAT. 

MOLECULAR 
WEIGHT. 

PRODUCT. 

n. 

C. 

Mercury  vapor  .     .".*.'..'*..' 

.015 

200 

3.00 

3 

1.00 

Hydrogen         

2.43 

2 

4.86 

5 

.97 

Oxvsren 

.154 

32 

4.93 

g 

.99 

Nitrogen  

.173 

28 

4.84 

5 

.97 

Carbon  monoxide  .... 

.173 

28 

4.84 

5 

.97 

Hydrochloric  acid  vapor     . 

.132 

36 

4.75 

5 

.95 

Water  vapor     

.371 

18 

6.68 

6 

1.11 

Carbon  dioxide     .... 

.15 

44 

6.60 

7 

.94 

The  average  of  these  values  of  C  is  0.98  ;  but  0.98  is  so  near 
to  1.00  that  the  data  that  we  have  are  not  sufficient  to 
distinguish  between  the  two  with  certainty.  We  may  there- 
fore consider  C  to  be  unity,  and  may  write  equation  (58)  thus  : 


sw  =  n. 


(59) 


Now  if  some  law  closely  analogous  to  Boltzmann's  holds  true 
for  solids,  then  Dulong  and  Petit's  law  might  follow  as  a 
natural  consequence.  If  we  strain  the  point  a  little  and 
apply  Boltzmann's  law  itself  to  solids,  we  see  that  the 
constancy  of  the  product  of  the  specific  heat  and  molecular 
weight  would  merely  imply  that  the  .  molecules  of  the 
substances  under  discussion  have  the  same  number  of  degrees 
of  freedom.  We  could  even  deduce  the  number  of  degrees  of 


158  THE   MOLECULAR    THEORY   OF   MATTER. 

freedom  in  any  given  case,  by  substituting  in  (59)  the  proper 
values  of  s  and  w,  remembering  (in  case  the  body  is  an 
element)  that  w  is  the  molecular  weight,  and  not  the  atomic 
weight.  The  elements  in  Meyer's  list  would  then  appear  to 
consist,  for  the  most  part,  of  molecules  having  11,  12,  or  13 
degrees  of  freedom.  Of  course  the  explanation  of  Dulong 
and  Petit's  law  that  I  am  offering  you  is  merely  speculative 
at  present ;  but  if  it  be  accepted  for  the  moment,  then  certain 
of  the  difficulties  that  beset  other  explanations  of  the  law 
disappear.  For  example,  it  is  probable,  beforehand,  that  the 
molecules  of  all  bodies  do  not  have  the  same  number  of 
degrees  of  freedom ;  and  hence,  in  accordance  with  (59),  we 
could  not  reasonably  expect  the  product  of  the  specific  heat 
and  the  molecular  weight  to  be  always  the  same,  but  should 
rather  expect  that  bodies  would  be  divisible  into  classes  in 
this  respect,  the  product  being  the  same  throughout  any  one 
class,  but  different  in  different  classes ;  and  this  is  the  fact, 
provided  the  law  be  extended  (as  I  think  it  should  be)  so  as 
to  include  compounds.*  Again,  we  can  understand  why  the 
law  is  not  exact ;  for  Boltzmann's  theorem  (and  presumably 
its  analogue  for  solids  also)  relates  only  to  the  partition  of 
kinetic  energy  among  the  various  degrees  of  freedom  of  the 
molecules.  The  law  of  distribution  of  potential  energy  is 
doubtless  quite  different ;  and  as  the  specific  heat  of  a  body 
corresponds  to  the  increase  in  its  total  energy,  it  follows  that 
(59)  will  not  be  strictly  true.  The  wide  divergence  from 
Dulong  and  Petit's  law  exhibited  by  certain  bodies  (such  as 
boron)  at  some  temperatures,  and  the  comparatively  good 
agreement  of  these  bodies  with  it  at  other  temperatures,  may 
be  due  either  to  a  change  in  the  number  of  degrees  of  freedom 
of  the  molecule,  or  to  a  local  variation  in  potential  energy 
great  enough  to  mask  the  effects  of  the  equable  distribution 
of  kinetic  energy. 

*  Of  course  numerous  attempts  to  so  extend  it  have  been  made  already  ; 
but  so  far  as  I  know,  they  have  not  been  based  on  the  considerations 
here  presented. 


PROUT'S  HYPOTHESIS.  159 

Prout's  Hypothesis.  —  The  idea  that  matter  is  not  really 
of  seventy  kinds  or  so,  but  that  it  consists  of  only  one  funda- 
mental kind,  is  quite  ancient ;  but  "  in  1815,  soon  after 
Dalton's  atomic  theory  had  met  with  general  recognition, 
Prout  brought  forward  the  view  that  the  primordial  matter 
of  which  all  elements  are  composed  is  hydrogen,  and  that 
consequently  the  atomic  weights  of  all  the  other  elements  are 
simple  multiples  of  the  atomic  weight"  of  that  substance.* 
This  hypothesis  has  provoked  much  discussion,  and  since  it 
was  first  proposed  it  has  been  attacked  and  defended  by  many 
distinguished  chemists ;  and  although  it  is  rather  in  disfavor 
at  present,  I  think  we  cannot  yet  say  that  it  has  been  finally 
laid  to  rest.  One  can  hardly  glance  at  a  table  of  atomic 
weights  without  being  impressed  by  the  close  approach  of 
these  quantities  to  integral  values.  Of  course  there  are 
conspicuous  exceptions  —  chlorine,  for  example,  —  to  Front's 
hypothesis  in  its  original  form,  and  to  reconcile  these  it  has 
been  assumed  that  the  various  elements  are  composed,  not  of 
hydrogen,  but  of  some  unknown  and  still  simpler  substance 
whose  atomic  weight  is  ^  or  1  that  of  hydrogen  ;  but  this 
seems  like  a  very  artificial  extension  of  the  hypothesis, 
because  by  a  further  extension  of  the  same  kind  we  could 
easily  account  for  any  exceptions  whatever.  The  fact  that 
many  of  the  atomic  weights  are  nearly  integral  demands  some 
sort  of  an  explanation,  however,  for  it  can  hardly  be  acci- 
dental. When  chemical  science  was  in  a  less  developed 
condition  it  was  easy  to  believe  that  the  atomic  weight  of 
nitrogen  (for  example)  is  14.00,  instead  of  14.02  as  indicated 
by  experiment,  and  that  the  atomic  weight  of  carbon  is  12.00 
instead  of  11.97 ;  but  we  can  110  longer  entertain  any  such 
hypothesis.  This  point  was  strongly  emphasized  by  Stas's 
magnificent  researches,  for  his  results  are  apparently  of  such 
extraordinary  accuracy  that  an  error  of  one-tenth  of  one  per 
cent  is  quite  out  of  the  question  in  them.  "  It  is  possible," 
says  Dr.  Lothar  Meyer,  "  that  the  atoms  of  all  or  many  of  the 

*  Meyer,  Modern  Theories  of  Chemistry,  page  113. 


160      THE  MOLECULAR  THEORY  OF  MATTER. 

elements  chiefly  consist  of  smaller  particles  of  matter  of  one 
distinct  primordial  form,  perhaps  hydrogen,  and  that  the 
weights  of  the  atoms  do  not  bear  a  simple  relation  to  one 
another  because  the  atoms  contain,  in  addition  to  the  particles 
of  this  primordial  matter,  varying  quantities  of  the  matter 
which  fills  space  and  is  known  as  the  luminif  erous  ether,  which 
is  perhaps  not  quite  devoid  of  weight.  This  appears  to  be 
the  only  permissible  hypothesis."  Dr.  Meyer's  surmise  may 
possibly  be  correct,  although  certain  grave  difficulties  would 
have  to  be  overcome  before  we  could  accept  it.  If  you  will 
bear  in  mind  what  I  said  a  few  moments  ago  about  all  these 
points  being  purely  speculative,  I  will  offer  another  hypothesis 
which  may  not  be  better  than  Dr.  Meyer's,  but  which  appears 
to  be  at  least  as  good,  and  quite  as  defensible.  There  is  no 
harm  in  letting  one's  fancy  loose  in  this  way,  any  more  than 
there  is  in  reading  a  fairy  tale  ;  but  it  is  of  the  first  impor- 
tance, in  either  case,  that  we  should  carefully  remember  what 
we  are  doing,  so  that  possibility  may  not  be  confused  with 
probability.  There  is  one  point  which  is  everywhere  taken 
to  be  self-evident  by  writers  on  chemistry,  but  which  is  not 
so,  to  me,  by  any  means.  I  cannot  see  what  warrant  there  is 
for  assuming  that  when  an  atom  whose  weight  is  A  combines 
with  another  atom  whose  weight  is  B,  the  weight  of  the 
resulting  molecule  is  universally  and  necessarily  A  +  B.  This 
principle,  instead  of  being  a  truism,  must  receive  a  most  exact 
explanation  by  the  final  molecular  theory.  It  appears  to  be 
true  in  such  reactions  as  we  can  observe,  but  as  we  have  never 
split  an  element  up  into  its  constituent  hydrogen-atoms  (if 
indeed  it  contains  such  atoms !)  there  is  no  evidence  that  in 
such  a  case  the  "  law  of  conservation  of  weight "  would  still 
hold  true.  When  we  know  more  about  the  nature  of  gravita- 
tion we  shall  be  in  a  better  position  to  discuss  this  point ;  but 
at  present  I  think  we  may  say  that  it  is  just  possible  that 
there  may  be  cases  in  which  an  atom  of  weight  A,  when  com- 
bining with  another  of  weight  B,  does  not  produce  a  molecule 
of  weight  A-\-B.  I  am  well  aware  that  this  would  make 


PERIODIC    LAW   OF   MEYER   AND   MENDELEIEFF.    161 

perpetual  motion  possible,  for  if  the  weight  of  the  given 
substances  happened  to  be  greater  in  the  combined  state  than 
in  the  uncombined  one,  we  should  only  have  to  let  them  fall 
some  convenient  distance  while  they  are  combined,  and  raise 
them  again  while  they  are  uncombined,  and  we  should  gain  a 
little  energy  every  time  the  cycle  was  repeated;  while  if 
combination  should  cause  a  loss  of  weight  instead  of  a  gain, 
we  could  attain  the  same  end  by  performing  the  cycle  in  the 
opposite  direction.  Now  I  am  sure  that  nobody  has  greater 
faith  in  the  conservation  of  energy  than  I  have,  and  yet  we 
should  remember  that  that  grand  principle,  the  discovery  of 
which  will  cause  the  nineteenth  century  to  be  remembered 
forever,  is  nevertheless  merely  an  abstraction  from  our 
experience;  and  that  it  teaches  us  nothing  except  that  we 
have  never  known  energy  to  be  created  or  destroyed,  and  that 
with  the  means  at  our  command  we  cannot  create  it  nor  destroy 
it.  If  it  be  true,  therefore,  that  matter  is  composed  of  some 
fundamental  substance  combined  with  itself  in  varying  degrees 
of  complexity,  then  whenever  the  law  of  conservation  of 
weight  would  be  violated  upon  splitting  a  body  up  into  its 
constituents,  or  in  forming  it  from  them,  the  means  at  our 
disposal  can  never  enable  us  to  effect  either  the  separation  or 
the  combination  ;  and  so  far  as  we  are  concerned,  such  a  body 
would  forever  remain  an  element.  On  the  other  hand,  when- 
ever the  law  of  conservation  of  weight  would  not  be  violated 
upon  splitting  a  body  up,  the  body  in  question  is  not  an 
element,  but  a  compound;  and  we  can  reasonably  hope  to 
effect  its  separation  into  two  or  more  simpler  bodies.  This 
hypothesis  explains  both  the  existence  of  "elements,"  and 
the  slight  deviations  from  integral  values  that  we  find  in 
their  atomic  weights.  I  offer  it  for  what  it  is  worth,  and 
have  nothing  further  to  say  in  defense  of  it. 

Periodic  Law  of  Meyer  and  Mendeleieff .  —  It  has  long 
been  known  that  certain  approximate  numerical  relations 
exist  among  the  atomic  weights  of  elements  that  have  similar 


162  THE   MOLECULAR   THEORY   OF   MATTER. 

properties.  For  example,  if  we  consider  the  three  closely 
related  elements  lithium,  sodium,  and  potassium,  we  find  that 
the  atomic  weight  of  sodium  is  almost  exactly  the  arithmetic 
mean  between  the  other  two.  Thus  we  have  Li  =  7.01  and 
K  =  39.02,  the  mean  of  which  is  23.02  ;  and  the  atomic 
weight  of  sodium  is  23.00.  Again,  in  the  triad  calcium, 
strontium,  and  barium,  we  have  Ca  =  39.99  and  Ba  =  136.76, 
the  mean  of  which  is  88.38  ;  and  the  atomic  weight  of 
strontium  is  87.37.  The  triad  chlorine,  bromine,  and  iodine, 
affords  another  familiar  example,  for  we  have  Cl  =  35.37  and 
I  =  126.56,  giving  a  mean  of  80.96 ;  and  the  atomic  weight  of 
bromine  is  79.77.  As  I  have  said,  these  peculiarities,  and 
numerous  other  similar  ones,  were  known  many  years  ago  ; 
but  we  are  still  ignorant  of  their  true  meaning.  In  recent 
years  a  great  multitude  of  relations  of  this  sort  among  the 
elements  have  been  brought  to  light,  and  two  distinguished 
chemists,  one  a  Russian  and  the  other  a  German,  have 
shown,  by  independent  investigations,  that  if  the  elements 
are  arranged  in  the  order  of  their  atomic  weights,  many  of 
their  properties  recur,  at  intervals,  in  a  sort  of  "periodic" 
manner,  as  we  pass  from  one  end  of  the  array  to  the  other. 
Tables  have  also  been  prepared,  in  which  the  elements  are 
arranged  in  rows  and  columns  in  such  a  manner  that  their 
relations  to  one  another  can  be  plainly  seen.  I  shall  not 
dwell  at  length  upon  this  "  periodic  law,"  because  an  adequate 
discussion  of  it  would  require  a  great  deal  of  time.  It  has 
been  set  forth  very  simply  and  clearly,  however,  and  also  in 
considerable  detail,  in  Meyer's  Modern  Theories  of  Chemistry, 
and  in  Mendeleieff's  Principles  of  Chemistry.  When  the 
attempt  was  made  to  arrange  the  elements  in  tabular  form, 
it  was  found  necessary  to  leave  certain  spaces  in  the  table 
vacant.  It  was  easy  to  imagine  that  elements  would  some 
day  be  found,  which  would  fit  into  these  spaces  ;  but  it  was 
much  more  difficult  to  predict  the  exact  properties  that  these 
hypothetical  elements  would  have.  Nevertheless,  Mendeleieff 
undertook  the  task,  and  in  several  instances  his  predictions 


PERIODIC    LAW    OF    MEYER    AND    MENDELEIEFF.    163 


have  been  fully  verified  by  subsequent  discovery.  One  of 
the  most  striking  instances  of  such  verification  is  afforded  by 
the  metal  known  as  scandium.  Mendeleieff's  predictions,  and 
the  actual  facts  as  they  were  afterwards  discovered,  are  here 
presented  in  parallel  columns,  and  you  will  see  that  the  corre- 
spondence is  extremely  close.  (The  unknown  element  was 
provisionally  called  "eka-boron,"  from  its  position  in  the 
table  of  elements.*) 


EKA-BORON  (hypothetical). 

1.  Atomic  weight  about  44. 

2.  Oxide  will  have  formula  Eb2- 
O3 ;  will  be  soluble  in  acids,  but  in- 
soluble in  alkalies  ;  specific  gravity 
about  3.5  ;  analogous  to  A1203,  but 
more  basic  ;  less  basic  than  MgO. 

3.  Salts  of  Eb  will  be  colorless, 
and  will  yield  gelatinous  precipi- 
tates with  KOH,  K2C03,  Na2HPO4, 
etc. 

4.  Sulphate  will  have  the  formu- 
la Eb2-3SO4,  and  will  form,  with 
K2S04,    a  double  salt  which  will 
probably  not  be  isomorphous  with 
the  alums. 


SCANDIUM  (actual). 

1.  Atomic  weight  =  44. 

2.  Oxide  has  formula  Sc203 ;  is 
soluble   in  strong  acids,    but   in- 
soluble in  alkalies;   specific  gravi- 
ty =  3.8  ;  analogous  to  A1203,  but 
more  decidedly  basic. 

3.  Solutions  of  Sc  salts  are  color- 
less, and  yield  gelatinous  precipi- 
tates with  KOH,  K2C03,  and  Na2- 
HP04. 

4.  Sulphate    has    the    formula 
Sc2  •  3  S04,    and    forms,    with    K2- 
S04,  the  double  salt  Sc2-3S04-3  K2- 
SO4,  which  is  not  an  alum. 


There  is  a  certain  analogy,  historically  at  least,  between 
this  "  periodic  law "  and  the  rough  arithmetical  progression 
known  to  astronomers  as  "  Bode's  law."  Bode's  law  success- 
fully predicted  the  asteroids,  and  assigned  them  their  place 
in  the  solar  system.  Afterwards,  when  Adams  and  Leverrier 
undertook  their  famous  labors  which  ended  in  the  discovery 
of  Neptune,  they  assumed,  naturally  enough,  that  the  unknown 
planet  would  also  conform  to  the  law  of  Bode,  and  they 
arranged  their  computations  accordingly.  When  the  planet 
had  been  discovered,  and  its  orbit  investigated,  it  was  found 
to  be  much  nearer  the  sun  than  had  been  anticipated.  In 


164  .   THE  MOLECULAR  THEORY  OF  MATTER. 

fact,  it  does  not  agree  with  Bode's  law  at  all,  and  the  "  law  " 
has  therefore  been  rejected,  and  is  now  regarded  only  as  a 
curiosity.  It  may  be  that  the  "  periodic  law  "  of  chemistry 
is  destined  to  a  like  fate,  through  the  discovery  of  new 
elements  that  cannot  be  placed  in  the  scheme  of  classification 
as  it  now  stands,  nor  in  any  modified  form  of  it ;  yet  this  is 
quite  improbable,  because  the  periodic  law  is  based  on  a  vast 
assemblage  of  facts  of  different  kinds  instead  of  upon  a  mere 
observed  arithmetical  progression,  and  also  because  the  law  is 
more  or  less  elastic,  as  will  be  evident  to  any  one  who  takes 
the  trouble  to  look  up  its  history  and  note  the  various  modifi- 
cations that  it  has  undergone  since  it  was  first  proposed. 
There  was  nothing  elastic  about  Bode's  law,  and  when  the 
adverse  fact  came,  the  "  law "  gave  way  before  it. 

Elastic-Solid  Theory  of  Light.  —  It  has  long  been  known 
that  light  is  not  propagated  with  infinite  rapidity,  but  that  it 
travels  with  a  finite  (though  prodigious)  speed.  It  has  also 
been  long  known  that  light  is  not  a  substance,  but  that  it  is  a 
mere  periodic  disturbance  of  some  kind  or  other.  Now  if  it 
be  admitted  that  light  is  a  disturbance  of  some  kind,  and  also 
that  it  can  travel  through  all  the  interstellar  spaces  of  the 
sidereal  universe,  it  follows  that  throughout  these  spaces  there 
must  be  some  body  that  is  disturbed;  for  it  would  be  highly 
absurd  to  suppose  that  a  disturbance  of  any  kind  could  take 
place  in  an  absolute  vacuum.  The  hypothetical  body  that  is 
thus  assumed  to  fill  all  space  is  called  the  luminiferous  (or 
"  light-bearing ")  ether.  Of  course  it  is  not  related  in  any 
way  to  the  volatile  liquid  that  chemists  know  as  "ether," 
and  it  is  unfortunate  that  the  same  word  should  be  applied  to 
such  different  things.  The  physicists  have  the  right  of  way 
here,  however,  for  their  ether  was  discovered  and  named  long 
before  chemists  produced  the  other  kind  of  ether.  Although 
the  ether  of  the  physicist  does  not  directly  affect  our  senses 
in  any  way,  and  is  quite  imperceptible  to  any  chemical  or 
physical  tests,  its  general  properties  have  nevertheless  been 


ELASTIC-SOLID    THEORY   OF    LIGHT.  165 

deduced  (by  making  certain  special  assumptions  about  the 
nature  of  light),  and,  as  might  be  expected,  they  have  been 
found  to  be  very  remarkable.  In  telling  you  of  them  I  will 
outline,  in  a  general  way,  what  is  known  as  the  "  elastic-solid  " 
theory  of  light ;  but  I  must  state,  before  doing  so,  that  this 
theory  has  been  recently  abandoned  and  replaced  by  another 
one  that  we  shall  consider  presently.  The  ether  being  assumed 
to  fill  all  space  (or  to  extend,  at  least,  to  the  most  remote 
visible  star),  it  was  conceived,  by  the  advocates  of  this  theory, 
to  be  thrown  into  a  state  of  vibration  by  the  violent  move- 
ments of  the  molecules  of  bodies.  If  the  ethereal  vibrations 
thus  set  up  were  comparatively  slow,  they  were  believed  to 
produce  the  phenomena  of  radiant  heat /  while  if  they  suc- 
ceeded one  another  rapidly  enough  to  affect  the  retina  of  the 
eye,  they  were  believed  to  constitute  light.  The  first  con- 
clusion with  regard  to  the  ether  was,  therefore,  that  it  is 
elastic  in  some  sense  or  other ;  because  an  inelastic  body  can- 
not transmit  vibrations.  Moreover,  since  many  of  the  stars 
that  are  visible  to  us  are  certainly  more  than  a  thousand  mil- 
lion million  miles  away,  there  was  good  reason  for  believing 
not  only  that  the  ether  is  elastic,  but  that  it  is  perfectly  so. 
Otherwise  the  light  of  the  fainter  stars  would  be  entirely 
extinguished  by  absorption  in  those  tremendous  wastes  of 
space.  Now  an  elastic  body  is  something  that  we  know  a 
good  deal  about,  from  experience  and  experiment ;  and  it  was 
therefore  easy  to  investigate  the  kinds  of  waves  that  such  a 
body  as  the  ether  can  transmit.  It  was  found  that  such 
waves  would  fall  into  two  general  classes.  In  the  first  place, 
if  the  ether  were  compressible  there  could  be  waves  of 
alternate  compression  and  rarefaction  ;  very  similar  to  the 
sound-waves  that  we  know  so  well  in  our  own  air.  Again, 
if  the  ether  could  sustain  a  shearing  stress,  as  solid  bodies 
can,  there  could  be  what  I  may  call  a  wave  of  propagation  of 
shearing  strain.  In  the  first  of  these  cases  the  to-and-fro 
displacements  of  the  ether-particles  would  take  place  in  a 
direction  parallel  to  the  direction  of  propagation  of  the  wave  ; 


166  THE   MOLECULAR   THEORY    OF   MATTER. 

and  in  the  second  case  the  displacements  would  be  perpen- 
dicular to  the  direction  of  propagation,  just  as  they  are  when 
an  undulation  travels  along  a  stretched  rope  or  wire.  Upon 
comparing  these  mathematical  results  with  the  facts  of 
nature,  it  was  easily  seen,  from  the  phenomena  of  polarized 
light,  that  waves  of  this  second  kind  do  actually  occur  in  the 
ether  ;  but  no  phenomena  could  be  discovered  which  could  be 
attributed  to  waves  of  the  first  kind,*  and  hence  it  was  con- 
cluded that  waves  of  the  first  kind  do  not  occur.  You  will 
see  that  the  reasoning  I  have  indicated  leads  to  the  strange 
results  (1)  that  the  ether  cannot  transmit  waves  of  com- 
pression, and  that  it  is  therefore  probably  absolutely  incom- 
pressible (I  say  probably,  because  the  non-existence  of  waves 
of  compression  and  rarefaction  could  also  be  explained  by 
merely  supposing  the  ether  to  be  devoid  of  elasticity  of 
volume),  and  (2)  that  it  can  sustain  a  shearing  stress,  and  is 
therefore  of  the  nature  of  an  elastic  solid.  Planets,  comets, 
and  even  such  tiny  things  as  atoms,  can  plunge  onward 
through  the  ether  without  experiencing  the  least  retarding 
effect  ;f  and  yet  certain  kinds  of  molecular  vibration  are 
picked  up  by  it  and  borne  away  with  prodigious  speed  into 
the  endless  depths  of  space.  The  non-resistance  offered  by  the 
ether  is  probably  due  to  its  incompressibility,  which  property 
prevents  the  establishment  of  a  wave  of  condensation  in  front 
of  the  body,  or  of  rarefaction  behind  it.  There  is  no  friction, 
and  no  eddies  are  produced.  The  great  velocity  of  ether- 
waves  (186,000  miles  a  second)  shows  either  that  the  density 
of  this  body  is  very  small,  or  that  its  rigidity  is  very  great. 
Attempts  have  been  made  to  determine  both  its  rigidity  and 

*  Unless  gravitation  is  such  a  phenomenon.  This  point  is  considered 
in  a  subsequent  section. 

t  The  retardation  experienced  by  Encke's  comet  has  often  been  attrib- 
uted to  the  resistance  of  the  ether  ;  but  as  this  retardation  has  been  only 
two-thirds  as  great,  since  1871,  as  it  was  in  former  years,  it  must  be 
referred  to  other  causes  —  probably  to  perturbations  from  some  unknown 
meteoric  stream. 


ELASTIC-SOLID   THEORY   OF   LIGHT.  167 

its  density,  and  although  no  accurate  results  have  been 
obtained,  certain  limits  have  been  assigned,  within  which 
they  are  likely  to  lie.  You  will  find  the  method  of  calcula- 
tion given  in  Maxwell's  article  on  Ether  in  the  Encyclopaedia 
Britannica.  The  minimum  limit  to  the  density,  as  obtained 
by  this  method,  is  5.36  X  10~19,  the  density  of  water  being 
unity.*  Although  the  ether  does  not  directly  retard  the 
motion  of  material  particles,  it  is  undoubtedly  influenced  by 
them  in  some  manner.  Thus  we  believe  that  it  penetrates  all 
bodies,  and  fills  up  the  spaces  between  their  molecules ;  and 
as  the  phenomena  of  refraction  show  that  the  velocity  of  light 
is  less  in  a  transparent  body  (say  in  glass)  than  it  is  in  a 
vacuum,  it  follows  that  the  ether  in  the  glass  has  either  a 
greater  density  or  a  less  rigidity  than  it  has  in  free  space. 
Either  of  these  suppositions  will  fit  the  case  under  considera- 
tion very  well ;  but  there  are  other  phenomena  that  will  not 
be  satisfied  so  easily,  and  it  has  been  found  to  be  impossible 
to  make  any  single  set  of  consistent  assumptions,  which  shall 
reconcile  the  "  elastic-solid "  theory  with  the  facts.  For 
example,  when  we  come  to  investigate  certain  problems  in 
partial  reflection  from  transparent  media,  and  others  relating 
to  diffraction  from  small  particles,  we  are  obliged  to  conclude 
that  it  is  the  density  of  the  ether  that  varies,  the  rigidity 
remaining  practically  constant.  On  the  other  hand,  the 
phenomena  of  double  refraction  require  us  to  admit  that  the 
rigidity  of  the  ether  in  a  doubly-refracting  body  is  different 
in  different  directions ;  and  hence  we  conclude  that  the 
rigidity  of  the  ether  is  altered  by  the  presence  of  molecules 
of  matter  —  a  conclusion  at  variance  with  that  previously 
reached  by  considering  the  phenomena  of  diffraction  and 
partial  reflection.  This  is  one  of  the  rocks  upon  which  the 

*  In  the  article  referred  to,  the  density  is  stated  to  be  9.36  X  10-19. 
This  result  is  erroneous  on  account  of  an  arithmetical  blunder,  as  any  one 
can  easily  see  by  trying  to  verify  Maxwell's  computation.  Of  course  the 
error  is  of  no  importance,  and  I  should  not  have  referred  to  it  had  I  not 
noticed  that  whenever  the  result  is  quoted,  the  erroneous  value  is  given. 


168  THE   MOLECULAR    THEORY    OF    MATTER. 

" elastic-solid"  theory  went  to  pieces.  There  are  numerous 
other  objections  to  it,  which  are  fully  as  serious  as  this  one'. 
For  example,  the  full  mathematical  theory  of  a  non-isotropic 
elastic  body  involves  the  consideration  of  no  less  than  twenty- 
one  coefficients ;  and  if  the  ether  within  a  doubly-refracting 
crystal  really  has  different  rigidities  in  different  directions, 
we  should  expect  the  phenomena  of  double-refraction  to  be 
much  more  complex  than  they  really  are.  In  other  words, 
the  "  elastic-solid  "  theory  is  too  ponderous.  It  tends  to  pre- 
dict things  that  do  not  exist,  and  in  order  to  prevent  it  from 
doing  so  we  have  to  make  certain  arbitrary  assumptions  about 
the  coefficients  that  occur  in  the  "  equations  of  motion  "  —  a 
proceeding  which  is  repugnant,  I  think,  to  every  philosophical 
mathematician.  Taking  into  account  the  various  difficulties 
that  have  arisen  in  the  course  of  its  development,  we  must 
admit,  with  Lord  Eayleigh,  that  "the  elastic-solid  theory, 
although  valuable  as  a  piece  of  purely  dynamical  reasoning, 
and  probably  not  without  mathematical  analogy  to  the  truth," 
is  no  longer  tenable.  If  we  dismiss  it  from  further  con- 
sideration as  being  incompetent  to  explain  the  entire  range 
of  optical  facts,  of  course  we  are  at  liberty  to  form  a  new 
conception  of  the  ether  also ;  for  the  properties  that  I  have 
already  assigned  to  this  body  were  merely  those  that  the 
"  elastic-solid "  theory  demanded ;  and  in  abandoning  that 
theory  we  abandon  all  its  consequences  at  the  same  time, 
and  prepare  ourselves  to  take  a  fresh  start  in  a  new  direc- 
tion—  which  direction,  fortunately,  has  already  been  pointed 
out. 

Electro -Magnetic  Theory  of  Light.  —  Many  years  ago 
Faraday,  discussing  the  supposed  phenomena  of  "  action  at  a 
distance  "  as  manifested  by  a  magnet,  said  that  he  believed 
that  there  is  some  mechanism  by  which  the  magnetic  influence 
is  enabled  to  extend  itself  through  a  space  apparently  vacuous. 
"  Such  an  action,"  he  said,  "  may  be  a  function  of  the  ether ; 
for  it  is  not  unlikely  that,  if  there  be  an  ether,  it  should  have 


ELECTRO-MAGNETIC    THEORY    OF    LIGHT.  169 

other  uses  than  simply  the  conveyance  of  radiation."  *  These 
are  profound  words  —  how  profound  they  are,  it  was  reserved 
for  Maxwell  to  show.  The  old  corpuscular  theory  of  light, 
defended  with  such  ingenuity  by  Sir  Isaac  Newton,  was 
thrown  overboard  long  ago  because  it  could  not  explain  the 
phenomena  of  interference,  and  for  other  reasons  that  I  need 
not  here  repeat.  Following  it  came  the  elastic-solid  theory, 
which  we  have  just  considered,  and  which  we  have  also  been 
obliged  to  abandon,  though  perhaps  with  some  reluctance. 
One  would  almost  be  ready  to  say  that  the  truth  must  lie  with 
one  of  these  two  theories,  for  it  would  appear  that  light  must 
be  either  some  kind  of  a  substance,  or  some  kind  of  a  motion 
in  a  substance.  Undoubtedly  this  is  true,  but  it  now  appears 
that  we  were  thinking  of  the  wrong  kind  of  a  motion, 
altogether.  Maxwell,  after  reading  Faraday's  experimental 
researches,  was  so  impressed  by  them  and  by  that  marvelous 
insight  into  things  which  seemed,  in  Faraday,  almost  like 
intuition,  applied  his  own  ingenious  and  powerful  mind  to  the 
problems  whose  solution  Faraday  had  dimly  glimpsed,  and 
succeeded  in  completely  re-volution izing  our  notions  of  light, 
and  showing  us  the  whole  subject  from  an  entirely  new  point 
of  view.  I  shall  not  attempt  to  discuss  the  general  theory  of 
electricity,  because  it  would  lead  us  too  far  away  from  our 
subject ;  but  I  must  indicate,  as  briefly  as  I  can,  the  nature 
of  Maxwell's  theory  of  light.  He  agrees  with  previous 
writers  that  light  is  some  sort  of  a  periodic  disturbance  in 
some  sort  of  an  ether,  and  that  the  displacements  that  occur 
as  the  wave  progresses  are  indeed  perpendicular  to  the 
direction  in  which  the  wave  travels ;  but  he  teaches  us  that 
these  displacements  are  not  analogous  to  those  that  are  pro- 
duced in  an  elastic-solid  when  that  solid  is  deformed.  He 
considers  that  they  are  of  an  electrical  nature,  and  that  we 
must  learn  about  them,  not  by  observing  the  behavior  of 
elastic  bodies  under  stress,  but  by  observing  the  phenomena 
exhibited  by  electrified  bodies  ;  and  this,  you  will  see,  is  an 
*  Experimental  Researches  in  Electricity,  Vol.  Ill,  p.  331. 


170      THE  MOLECULAR  THEORY  OF  MATTER. 

entire  change  of  base.  Maxwell  has  given  us  the  funda- 
mental equations  that  must  be  satisfied  when  an  electrical 
wave  is  propagated  through  the  ether  —  equations  analogous 
to  the  "  equations  of  motion "  of  the  old  elastic-solid  theory 
—  and  by  means  of  these  equations  the  entire  theory  of  light 
can  be  constructed  on  the  new  basis.  The  theory  thus  con- 
structed agrees  well  with  the  facts  of  observation,  and  it  is 
free  from  the  numerous  objections  that  beset  the  old  elastic- 
solid  theory.  Moreover,  it  successfully  withstood  the  search- 
ing experimental  tests  devised  by  the  late  Professor  Hertz, 
whose  labors  have  shown  us  in  a  very  direct  manner  that 
electrical  radiations  are  propagated  with  the  same  speed  as 
light,  and  that  they  can  be  reflected,  refracted,  diffracted, 
polarized,  and  made  to  interfere ;  so  that  we  are  now  quite 
ready  to  admit  that  light  consists  in  a  rapid  succession  of 
such  radiations.  It  is  not  at  all  essential  to  Maxwell's  theory 
of  light  that  we  should  know  what  an  "  electrical  displace- 
ment" really  is.  We  derive  his  fundamental  equations  from 
a  study  of  electrical  phenomena  as  observed  in  gross  matter, 
and  we  then  apply  these  equations  to  the  ether,  and  deduce, 
by  means  of  them,  the  laws  that  govern  the  propagation  of 
electrical  disturbances  in  that  body.  We  then  obsej've  that 
the  laws  so  deduced  are  precisely  the  same  as  those  that  have 
long  been  known  to  hold  true  for  light ;  and  hence  we  con- 
clude that  light  is  an  electrical  phenomenon.  This  is  the 
whole  story,  so  far  as  we  have  any  positive  knowledge  of  it  at 
present.  We  have  some  dim  ideas  about  the  nature  of  electric 
and  magnetic  displacements,  but  I  think  it  is  safe  to  say  that 
we  know  little  or  nothing  about  them  that  is  not  liable  to  be 
profoundly  modified  by  subsequent  research.  It  is  quite 
probable  that  there  is  some  kind  of  an  ethereal  rotation  going 
on  in  a  magnetic  field,  because  it  is  hard  to  account  for 
magnetic  rotation  of  the  plane  of  polarization  on  any  other 
hypothesis.  However,  it  should  be  remembered  that  even 
this  assumption  is  by  no  means  beyond  controversy,  for  the 
plane  of  polarization  of  light  is  affected  by  magnetism  only 


PROVISIONAL  ASSUMPTIONS.  171 

when  molecules  of  gross  matter  are  present.  There  is  a  great 
deal  of  work  to  be  done  before  we  can  form  a  clear  and  true 
conception  of  the  ether,  and  of  what  goes  on  in  it  in  the 
vicinity  of  an  electrified  body,  or  a  magnet,  or  a  ray  of  light ;  * 
but  when  such  a  conception  has  been  attained,  we  shall  inci- 
dentally learn  a  great  deal  about  the  relation  of  ether  to 
matter,  and  about  the  constitution  of  the  molecules  of  which 
matter  is  composed. 

Provisional  Assumptions  about  the  Constitution  of  Mole- 
cules. —  In  discussing  the  molecular  theory  of  matter  I  have 
made  certain  assumptions  about  the  constitution  of  molecules, 
which  are  perhaps  the  most  natural  ones  to  make,  and  which 
ought  therefore  to  serve  as  a  basis  for  our  investigations,  at 
least  until  it  appears  that  they  are  inadequate,  or  that  some 
other  assumptions  would  be  better.  Thus  I  have  assumed 
that  molecules  are  composed  of  smaller  bodies  called  atoms, 
which  are  held  together  (but  not  necessarily  in  contact)  by 
certain  attractive  forces,  whose  precise  nature  we  have  not 
determined.  The  atoms  have^  been  assumed,  furthermore,  to 
have  definite  forms  and  sizes,  to  be  perfectly  elastic,  and  to 
be  subject  to  the  same  laws  of  mechanics  that  govern  the, 
larger  masses  of  matter  that  we  can  observe  directly.  These 
assumptions  are  simple  enough,  but  it  is  far  from  certain  that 
they  correspond  with  the  facts  ;  and  at  all  events  they  are 
open  to  certain  grave  philosophical  objections  which  we  shall 
presently  consider,  in  connection  with  Lord  Kelvin's  vortex- 
theory.  The  attractive  forces  that  undoubtedly  exist  between 
molecules  must  receive  some  kind  of  a  mechanical  explanation, 
but  although  some  attempts  have  been  made  to  provide  such 

*  Dr.  Oliver  J.  Lodge's  Modern  Views  of  Electricity  (Macmillan  &  Co., 
1889)  is  an  excellent  book  on  this  subject,  though  I  fear  that  his  ingen- 
ious models  tend  to  give  students  a  too  mechanical  conception  of  the 
ether  and  of  electric  action,  and  to  deceive  them  into  the  belief  that  we 
know  much  more  about  these  things  than  we  really  do.  It  is  possible, 
however,  that  I  am  too  conservative  on  this  point. 


172  THE   MOLECULAR    THEORY    OF    MATTER. 

an  explanation,  the  subject  is  still  quite  obscure.*  These 
forces  are  sometimes  thought  to  be  of  an  electrical  nature ; 
but  until  we  know  more  about  the  ether  and  the  mechanism 
of  electric  attraction  such  an  assumption  can  hardly  be  con- 
sidered satisfactory.  The  elasticity  of  molecules  certainly 
admits  of  explanation,  and  something  has  already  been  done 
in  this  direction,  as  we  shall  presently  see.  It  is  far  from 
certain  that  molecules  even  have  definite  dimensions  ;  for  they 
may  be  mere  centers  of  condensation  of  the  ether,  or  of  some 
non-ethereal  substance  distributed  through  it,  and  they  may 
be  as  indefinite  in  their  boundaries  as  the  nebulae  that  we  see 
in  the  heavens.  We  do  not  even  know  that  those  general 
principles  of  mechanics  that  we  call  "axioms,"  and  which 
are  derived  from  our  observation  of  vast  aggregates  of  mole- 
cules, are  applicable,  without  modification,  to  the  molecules 
themselves  ;  and  yet  I  think  it  is  logical  for  us  to  adopt  them 
until  it  can  be  shown  that  they  lead  to  false  results.  So  far 
as  the  general  assumptions  that  I  have  made  this  evening  are 
concerned,  I  think  it  can  be  said  that  there  is  no  gross  failure 
in  the  molecular  theory  that  can  be  attributed  to  them.  The 
present  fragmentary  state  of  the  theory  appears  to  be  attrib- 
utable to  the  enormous  mathematical  difficulties  that  are 
involved,  rather  than  to  error  in  the  premises.  As  I  have 
told  you,  the  assumptions  that  we  have  made  are  liable  to 
certain  philosophical  objections,  and  for  this  reason  we  must 
consider  them  to  be  merely  provisional,  accepted  for  the 
present  as  convenient  stepping  stones,  but  subject  to  revision 
as  the  growth  of  the  molecular  theory  proceeds.  We  are  not 
yet  prepared  to  develop  the  molecular  theory  from  any  point 
of  view  that  differs  materially  from  that  which  I  have  given 
you,  but  it  will  be  interesting  to  note  the  direction  in  which 
future  research  is  likely  to  lead  us,  and  for  this  reason  I  must 
notice  two  or  three  of  the  more  important  hypotheses  that 
have  been  advanced,  concerning  the  constitution  of  molecules. 

*  Some  of  these  attempts  are  considered  in  a  subsequent  section  on 
"Gravitation." 


RANKINE'S    HYPOTHESIS.  173 

Rankine's  Hypothesis.  —  I  do  not  quite  know  what  I  ought 
to  say  about  Eankine's  views  concerning  the  constitution  of 
molecules.  He  certainly  did  deduce  many  of  the  known 
properties  of  bodies  from  his  "hypothesis  of  molecular 
vortices,"  but  I  am  not  aware  that  any  other  mathematician 
has  found  that  hypothesis  promising  enough  to  call  for 
further  investigation.  He  attributes  the  hypothesis  to  Sir 
Humphrey  Davy,  but  it  has  long  been  known  by  Eankine's 
name  because  he  was  the  first  to  develop  it  by  mathematical 
methods.  The  hypothesis  of  molecular  vortices  assumes 
"that  each  atom  of  matter  consists  of  a  nucleus  or  central 
point  enveloped  by  an  elastic  atmosphere,  which  is  retained 
in  its  position  by  attractive  forces,  and  that  the  elasticity  due 
to  heat  arises  from  the  centrifugal  force  of  those  atmospheres, 
revolving  or  oscillating  about  their  nuclei  or  central  points."* 
He  does"  not  attempt  to  decide  "  whether  the  elastic  atmos- 
pheres are  continuous,  or  consist  of  discrete  particles";  nor 
did  he  find  it  necessary  to  determine  whether  the  nucleus  of 
a  molecule  "  is  a  real  nucleus  having  a  nature  distinct  from 
that  of  the  atmosphere,  or  a  portion  of  the  atmosphere  in  a 
highly  condensed  state,  or  merely  a  center  of  condensation  of 
the  atmosphere,  and  of  resultant  attractive  and  repulsive 
forces."  He  believed  "that  the  vibration  which,  according 
to  the  undulatory  hypothesis,  constitutes  radiant  light  and 
heat,  is  a  motion  of  the  atomic  nuclei  or  centers,  and  is 
propagated  by  means  of  their  mutual  attractions  and  repul- 
sions." This  form  of  the  theory  of  light  receives  some 
considerable  support  from  the  phenomena  of  double  refraction 
and  polarization,  but  of  course  it  has  no  bearing  on  the  mode 
of  propagation  of  light  through  free  space,  unless  the  ether 
itself  is  conceived  to  have  a  similar  constitution.  Eankine 
perceived  this  fact  very  clearly,  and  in  an  interesting  paper 
on  light,  read  before  the  British  Association  in  1853,t  he 

*  Rankine,  Miscellaneous  Scientific  Papers  (London,  Charles  Griffin. 
&Co.,  1881),  page  17. 

t  Miscellaneous  Scientific  Papers,  page  156. 


174  THE   MOLECULAR    THEORY    OF    MATTER. 

assumes  "that  the  luminiferous  medium  is  composed  of 
detached  atoms  or  nuclei  distributed  throughout  all  space, 
and  endowed  with  a  peculiar  species  of  polarity,  in  virtue  of 
which  three  orthogonal  axes  in  each  atom  tend  to  place  them- 
selves parallel  respectively  to  the  corresponding  axes  in  every 
other  atom  ;  and  that  plane-polarized  light  consists 'in  a  small 
oscillatory  movement  of  each  atom  round  an  axis  transverse 
to  the  direction  of  propagation."  A  serious  objection  to 
Eankine's  hypothesis  of  molecular  vortices  is,  that  it  seems 
to  hold  out  little  promise  of  eventually  offering  a  mechanical 
explanation  of  attraction.  He  merely  postulates  the  attrac- 
tion, and  when  we  look  for  some  sufficient  cause  for  it,  the 
hypothesis  is  barren,  and  its  prophet  dumb.  The  facts  of 
chemistry  are  also  hard  to  explain  from  Rankine's  point  of 
view ;  and  if  we  judge  the  hypothesis  of  molecular  vortices 
according  to  its  fruits,  we  must  pronounce  it  a  step  in  the 
wrong  direction ;  for  it  still  remains  where  it  was  some  forty 
years  ago. 

Lord  Kelvin's  Vortex  Theory.  — We  have  now  to  consider 
a  very  curious  and  interesting  theory  of  the  constitution  of 
molecules,  which  was  originally  proposed  by  Lord  Kelvin. 
Most  of  the  theories  that  have  been  advanced  have  assumed 
that  there  are  two  kinds  of  matter,  one  being  that  which  we 
ordinarily  call  "  matter,"  and  the  other  being  the  imponderable 
"  ether,"  whose  existence  we  have  been  obliged  to  admit  in 
order  to  account  for  the  phenomena  of  electricity,  light,  and 
radiant  heat.  Lord  Kelvin  has  dispensed  with  one  of  these 
substances  altogether,  by  assuming  that  molecules  (or  the 
atoms  composing  them)  are  merely  definite  portions  of  the 
ether  itself,  which  are  distinguished  from  the  remainder  of 
that  vast  body  by  being  endowed  with  a  peculiar  kind  of 
motion,  called  "  vortex  motion."  I  should  like  to  give  you  a 
•clear  idea  of  vortex  motion,  and  perhaps  it  will  be  best  for  me 
to  begin  by  calling  attention  to  certain  cases  of  it  that  you  are 
likely  to  have  seen.  The  grandest  example  occurring  in 


LORD  KELVIN'S  VORTEX  THEORY.  175 

nature  is  the  cyclone,  which  consists,  as  you  know,  in  a  violent 
rotation  of  the  air  about  a  central  vertical  axis,  accompanied 
by  a  translation  of  the  axis  in  a  direction  perpendicular  to  its 
length.  You  have  probably  often  seen  smoke-rings  blown 
from  the  stack  of  a  locomotive,  and  if  you  have  observed 
them  closely  you  have  noticed  that  they  are  in  rapid  rotation, 
the  smoke-particles  passing  up  through  the  ring  on  the  inside 
and  down  again  on  the  outside.  These  rings  are  good 
examples  of  vortex  motion  in  which  the  axis  of  rotation 
returns  into  itself  so  as  to  form  a  closed  curve.  Experienced 
smokers  can  often  produce  similar  rings,  on  a  small  scale, 
with  their  lips  ;  and  very  good  ones  can  be  made  by  the 
simple  apparatus  devised  by  Professor  Tait  and  shown  in 
Fig.  50.  This  apparatus  consists  of  a  box  with  a  round  hole 
several  inches  in  diameter  in  one  end  of  it,  the  other  end 
being  removed  and  replaced  by  a  tense  sheet  of  india  rubber. 


B'  A 


FIG.  50.  —  TAIT'S  APPARATUS  FOR  PRODUCING 
SMOKE-RINGS. 


In  order  to  make  the  rings  visible,  the  box  may  be  filled  with 
smoke  or  some  similar  substance.  Professor  Tait,  for  this 
purpose,  makes  use  of  the  dense  cloud  of  sal  ammoniac 
particles  that  is  produced  when  vapors  of  ammonia  and 
hydrochloric  acid  are  allowed  to  mingle  with  each  other. 
The  ammonia  is  sprinkled  over  the  inside  of  the  box,  and  the 
hydrochloric  acid  is  generated  by  pouring  strong  sulphuric 
acid  over  some  salt  contained  in  a  saucer  which  rests  upon 
the  bottom  of  the  box.  If  the  stretched  sheet  of  rubber  be 
now  gently  struck,  a  beautiful  smoke-ring  issues  from  the 
front  of  the  box.  The  constitution  of  the  smoke-rings  pro- 
duced in  this  manner  is  indicated  in  Fig.  51,  where  the  small, 


176      THE  MOLECULAR  THEORY  OF  MATTER. 

curved  arrows  indicate  the  direction  of  the  rotation,  and  the 
large,  straight  one  shows  the  direction  in 
which  the  ring  travels.*  A  great  variety 
of  beautiful  experiments  may  be  tried  by 
means  of  this  simple  apparatus.  For 
example,  we  may  study  the  action  between 
a  pair  of  rings  by  producing  two  of  them 
in  quick  succession,  as  shown  at  A  and  B, 
in  Fig.  50.  When  the  experiment  is  suc- 

FiG.51.  — DIAGRAM  OF  cessful  we  shall  see  the  first  ring  enlarge 
A  SMOKE-RING.  and  siac^en  jts  Speed  of  translation,  while 

the  second  one  grows  smaller  and  moves  faster,  so  that  it 
presently  passes  through  the  first  one,  and  the  rings  take  the 
relative  positions  indicated  at  B '  and  A'.  The  ring  A '  being 
now  the  foremost  one,  tends  to  slow  down  and  enlarge,  and 
the  ring  B '  tends  to  grow  smaller  and  move  faster  so  as  to 
pass  through  A',  and  so  on  perpetually ;  but  it  is  difficult  to 
realize  more  than  one  such  passage,  in  the  actual  experiment, 
because  the  viscosity  of  the  air  soon  stops  the  rotation  of  the 
ring,  and  when  the  rotation  has  ceased  the  ring  is  no  longer 
a  vortex,  but  merely  a  wreath  of  smoke.  Helmholtz  was  the 
first  man  to  investigate  vortex  motion  by  rigid  mathematical 
methods,  and  some  of  his  results  are  very  interesting.  In  his 
researches  the  fluid  in  which  the  vortices  exist  was  assumed 
to  be  frictionless,  homogeneous,  and  incompressible.  These 
properties  being  admitted,  he  showed  (1)  that  a  vortex  can 
never  be  produced  nor  destroyed  in  a  medium  of  this  character, 
so  that  if  such  vortices  exist,  they  will  continue  to  exist  for- 
ever ;  (2)  that  a  vortex  cannot  have  a  free  end  within  the 
fluid,  and  hence  every  vortex  must  either  return  into  itself  so 
as  to  form  a  closed  curve  (like  a  smoke-ring),  or  be  infinite  in 

*  It  seems  hardly  necessary  to  say  that  the  smoke  and  the  sal  ammoniac 
fumes  play  no  part  whatever  in  these  experiments,  except  to  make  the 
rings  visible.  The  existence  of  the  rings  can  be  demonstrated  when  the 
smoke  is  entirely  absent,  by  the  effects  produced  on  a  distant  sheet  of 
tissue  paper,  or  a  candle-flame. 


LORD  KELVIN'S  VORTEX  THEORY.  177 

length,  or  have  its  ends  upon  a  bounding  surface  of  the  fluid ; 
(3)  that  a  vortex  always  consists  of  the  same  portion  of  fluid, 
so  that  when  it  travels  through  the  surrounding  fluid  it  is  not 
alone  the  motion  which  progresses  (as  would  be  the  case  in  a 
wave)  ;  the  vortex  does  not  lose  its  identity,  but  "  moves  "  in 
the  same  sense  that  a  projectile  moves  when  propelled  through 
the  air;  (4)  no  two  vortices  can  ever  intersect  each  other, 
and  no  vortex  can  ever  intersect  itself.  It  is  also  known  that 
a  vortex  in  a  frictionless,  incompressible  fluid  would  behave 
like  a  perfectly  elastic  body.  Many  other  properties  of 
vortices  have  been  deduced,  but  those  that  I  have  mentioned 
will  be  sufficient  for  our  present  purposes.  Some  eight  years 
after  the  publication  of  Helmholtz's  paper  on  vortex  motion, 
and  while  watching  Professor  Tait's  beautiful  experiments  on 
smoke-rings,  Lord  Kelvin  conceived  the  idea  that  atoms  may 
possibly  be  vortices  in  the  luminiferous  ether.  There  is  much 
to  be  said  in  favor  of  this  hypothesis,  from  a  philosophic 
point  of  view.  As  we  have  seen,  it  enables  us  to  dispense 
with  one  of  the  two  "  kinds  of  matter  "  entirely,  for  it  teaches 
that  all  things  are  composed  primarily  of  ether,  and  that 
"  gross  matter  "  is  distinguished  from  the  surrounding  medium 
solely  by  its  being  endowed  with  the  peculiar  kind  of  motion 
that  we  have  just  been  considering.  It  explains  the  perma- 
nence of  "gross  matter,"  because  Helmholtz's  investigations 
prove  that  a  vortex-atom  in  a  frictionless  fluid  can  never  be 
created  nor  destroyed.  It  explains  the  elasticity  of  molecules, 
because  it  shows  that  an  ether-vortex  would  behave  like  a 
perfectly  elastic  body,  even  though  the  ether  itself  were 
entirely  devoid  of  elasticity.  Most  of  the  theories  of  matter 
leave  the  existence  of  some  70  or  so  elements  as  much  of  a 
mystery  as  the  existence  of  "  species  "  was,  in  the  biological 
world,  before  the  time  of  Spencer  and  Darwin ;  but  the 
vortex-theory  gives  us  at  least  a  suggestion  on  this  point.  I 
have  already  told  you  that  a  finite  vortex,  in  an  infinite, 
frictionless  fluid,  must  return  into  itself;  but  there  is  no 
reason  for  assuming  that  it  must  form  a  simple  ring,  like  a 


178  THE    MOLECULAR    THEORY    OF    MATTER. 

smoke-ring.  There  is  no  assignable  reason  why  it  could  not 
have  the  form  suggested  in  Fig.  52,  or  any  other  more  compli- 
cated form  ;  and  since  a  vortex  can  never  intersect  itself,  the 
degree  of  knottedness  of  a  vortex-atom 
can  never  change.  It  may  be,  there- 
fore, that  the  elements  differ  in  their 
properties,  on  account  of  their  atoms 
differing  in  knottedness.  The  vortex- 
theory  of  Lord  Kelvin  also  holds  out 

FiG~"52.  —  A  KNOTTED        some    faint    promise    of    explaining 
VORTEX.  other  facts  of  chemistry ;  and  in  this 

respect,  at  least,  it  is  decidedly  superior  to  Eankine's  hypothe- 
sis. I  will  not  attempt  to  say  just  what  explanation  of 
chemical  combination  might  prove  to  be  best,  but  there  is  a 
certain  suggestiveness  in  the  behavior  of  a  pair  of  similar  and 
nearly  parallel  smoke-rings,  which  tend  to  thread  through  and 
through  each  other  perpetually,  as  illustrated  in  Fig.  50.  A 
host  of  other  possibilities  lie  before  the  vortex-theory,  but  it 
is  doubtful  if  further  speculation  would  be  profitable  for  us. 
The  consequences  of  the  vortex-theory  can  be  deduced  by 
rigid  mathematical  methods,  and  it  is  idle  to  try  and  guess 
them  in  advance.  In  fact,  one  of  the  greatest  philosophical 
advantages  of  the  vortex-theory  is,  that  it  admits  of  so  few 
assumptions.  Other  theories  are  more  or  less  elastic,  and  can 
be  modified  so  as  to  bring  them  into  harmony  with  each  new 
phenomenon  ;  but  when  the  fundamental  assumptions  of  the 
vortex-theory  have  once  been  made,  we  are  bound  to  adhere 
to  them,  and  to  deduce  from  them,  by  exact  analysis,  all  the 
known  properties  of  matter.  As  Maxwell  says,  "  When  the 
vortex-atom  is  once  set  in  motion,  all  its  properties  are  abso- 
lutely fixed  and  determined  by  the  laws  of  motion  of  the 
primitive  fluid,  which  are  fully  expressed  in  the  fundamental 
equations.  The  disciple  of  Lucretius  may  cut  and  carve  his 
solid  atoms  in  the  hope  of  getting  them  to  combine  into 
worlds ;  the  follower  of  Boscovich  may  imagine  new  laws  of 
force  to  meet  the  requirements  of  each  new  phenomenon  ;  but 


179 

he  who  dares  to  plant  his  feet  in  the  path  opened  up  by 
Helmholtz  and  Thomson  [Lord  Kelvin]  has  no  such  resources. 
His  primitive  fluid  has  no  other  properties  than  inertia, 
invariable  density,  and  perfect  mobility,  and  the  method  by 
which  the  motion  of  this  fluid  is  to  be  traced  is  pure  mathe- 
matical analysis.  The  difficulties  of  this  method  are  enor- 
mous, but  the  glory  of  surmounting  them  would  be  unique."* 
The  vortex-theory  is  inseparably  united  to  the  theory  of 
electricity  and  light,  since  both  of  these  theories  involve  a 
discussion  of  the  ether ;  and  it  remains  to  be  seen  whether  a 
constitution  can  be  imagined  for  that  body  which  shall 
explain  the  propagation  of  radiant  energy,  without  excluding 
the  possibility  of  vortex-motion.  Before  the  old  elastic-solid 
theory  of  light  was  abandoned,  the  vortex-atom  could  hardly 
be  seriously  considered ;  for  a  vortex  in  an  elastic-solid  is  a 
manifest  absurdity.  The  electro-magnetic  theory  cleared  the 
way  for  the  vortex-atom,  however,  by  teaching  us  that  our 
elastic-solid  analogy  was  erroneous.  We  are  now  free  to  form 
a  new  conception  of  the  ether,  which  may  possibly  reconcile 
the  vortex-atom  with  the  theory  of  light ;  but  our  past 
experience  in  this  direction  has  shown  us  that  we  should 
proceed  with  extreme  caution.  The  advocates  of  the  vortex- 
theory  are  extending  their  theory  to  the  ether  itself,  in  an 
attempt  to  explain  how  that  body  may  be  a  frictionless, 
incompressible  fluid,  and  yet  have  elasticity.  For  this 
purpose  the  ether  is  regarded  as  a  perfect  snarl  of  minute, 
interlacing  vortices,  which  are  normally  in  equilibrium,  but 
which  serve  as  an  elastic  framework  for  the  transmission  of 
radiant  energy.  This  branch  of  the  vortex-theory  is  too 
abstruse  and  too  imperfectly  developed  to  be  considered 
further  in  this  place ;  and  it  may  be  said  of  the  vortex-theory 
in  general,  that  it  is  full  of  enormous  mathematical  difficulties, 
and  that  for  this  reason  we  can  regard  it,  at  present,  only  as 
a  highly  interesting  possibility,  whose  consequences  must  be 
traced  out  by  future  generations. 

*  Encyclopaedia  Britannica,  article  Atom. 


180     THE  MOLECULAR  THEORY  OF  MATTER. 

Dr.  Burton's  Strain-Figure  Theory.  —  Numerous  other 
theories  of  the  constitution  of  molecules  have  been  advanced, 
but  most  of  them  are  open  to  so  many  objections  that  they 
cannot  be  considered  to  be  tenable  at  present,  and  need  not 
be  discussed  in  this  place.  As  an  example  I  may  mention 
the  complicated  theory  of  Lindemann,  which  considers  a 
molecule  to  consist  of  a  series  of  concentric  spherical  shells, 
each  of  which  is  elastically  connected  with  its  neighbors.  Dr. 
C.  V.  Burton's  theory  must  be  briefly  mentioned,  however,  for 
although  it  is  in  a  very  imperfectly  developed  condition,  it 
presents  points  of  novelty  that  cannot  fail  to  broaden  our 
conception  of  what  a  molecule  may  be.*  His  theory  bears  a 
superficial  resemblance  to  Lord  Kelvin's,  inasmuch  as  it  con- 
siders an  atom  to  consist  of  a  modified  portion  of  the  ether ; 
but  further  than  this  the  two  theories  are  radically  different. 
Dr.  Burton  conceives  that  the  ether,  although  possibly  of  a 
fluid  nature,  is  nevertheless  endowed  with  some  sort  of 
elasticity  (which  is  no  doubt  the  case,  since  otherwise  it 
could  not  transmit  radiant  energy).  He  further  assumes  that 
small  strains  in  the  ether  are  always  proportional  to  the 
stresses  that  accompany  them ;  but  that  when  the  strains 
exceed  a  certain  limit  the  ether  takes  a  sort  of  "  permanent 
set,"  after  which  it  never  returns  to  its  primitive  condition. 
Dr.  Burton  assumes  that  atoms  are  merely  deformations  in 
the  ether  that  have  been  produced  by  such  a  process  of  over- 
straining. Lord  Kelvin's  vortex-theory  has  been  facetiously 
called  the  "  doughnut-theory,"  and  perhaps  we  may  designate 
Dr.  Burton's  theory,  in  the  same  spirit,  as  the  "ether-dent 
theory  " ;  though  neither  term  is  very  apt,  for  a  vortex  is  not 
necessarily  a  simple  ring,  and  a  "  strain-figure  "  in  the  ether 
is  far  from  being  a  mere  dent.  To  give  you  a  clearer  idea  of 
Dr.  Burton's  fundamental  assumption  I  will  quote  his  own 
description  of  it.  (i Consider  a  region,"  he  says,  "either 

*  For  a  full  account  of  Dr.  Burton's  theory  see  his  paper  entitled 
A  Theory  Concerning  the  Constitution  of  Matter,  in  the  Philosophical 
Magazine  for  February,  1892. 


DR.  BURTON'S  STRAIN-FIGURE  THEORY.         181 

infinite  or  having  very  distant  boundaries,  and  filled  with 
a  homogeneous  isotropic  elastic  medium,  whose  condition 
throughout  is  one  of  stable  equilibrium  for  small  strains  of 
any  type.  Let  the  medium  now  be  strained,  and  held  in  its 
strained  condition  by  some  compelling  agency :  there  will  be 
a  corresponding  distribution  of  stress  in  the  medium,  and, 
provided  the  strain  has  at  no  point  too  great  a  value,  the 
original  condition  will  be  completely  regained  after  the  com- 
pelling agency  has  been  removed.  But  suppose  that,  instead, 
the  medium  is  strained  further  and  further  from  its  initial 
state,  and  suppose  that  the  restoring  stresses  do  not  always 
increase  with  the  strain,  but  that  beyond  a  certain  point  in 
the  process  they  begin  to  fall  off  in  value,  until  at  last  a  point 
is  reached  at  which  the  general  tendency  of  the  stress  is  to 
further  increase  the  strain.  If  the  compelling  agency  is  now 
withdrawn,  the  medium  will  subside  into  a  new  condition  of 
stable  equilibrium,  involving  stress  and  strain  at  every  point. 
The  state  of  things  thus  impressed  on  the  medium  is,  accord- 
ing to  my  view,  an  atom  or  a  constituent  of  an  atom ;  it  will 
hereafter  be  referred  to  as  a  strain-figure"  This  passage  does 
not  purport  to  explain  the  origin  of  matter  ;  it  is  intended 
merely  to  convey  to  the  reader  the  meaning  of  the  term 
"  strain-figure "  as  used  in  Dr.  Burton's  paper.  It  is  sug- 
gested, however,  that  if  the  ether  "  had  long  ago  possessed 
motion  of  the  most  general  kind,  we  might  imagine  its  present 
condition  to  be  due  to  the  degeneration  of  that  motion  into  a 
fine-grained  turbulence ;  and  if,  in  the  quasi-solid  so  consti- 
tuted, the  existence  of  strain-figures  were  possible,  it  seems 
not  unlikely  that  such  would  incidentally  have  been  formed, 
unless  the  motion  fulfilled  special  conditions."  If  such 
special  conditions  were  absent,  it  is  therefore  possible,  on  the 
strain-figure  hypothesis,  that  atoms  would  have  resulted,  from 
time  to  time,  whenever  and  wherever  the  motion  of  the  ether 
should  chance  to  be  such  as  to  produce  a  strain  sufficient  to 
give  rise  to  a  "permanent  set."  Dr.  Burton's  theory,  there- 
fore,, holds  out  some  hope  of  even  explaining  the  origin  of 


182      THE  MOLECULAR  THEORY  OF  MATTER. 

matter ;  and  in  this  respect  it  differs  from  every  other  theory 
with  which  I  am  familiar.  A  mathematical  analysis  of 
strain-figures  shows  that  they  would  possess  many  of  the 
characteristics  that  atoms  are  supposed  to  have.  They  could 
be  moved  about  in  the  ether  without  encountering  resistance ; 
but  we  are  to  consider  that  when  such  motion  occurs  it  is 
not  the  ether  itself  that  moves,  but  that  the  modification  of 
structure  that  constitutes  a  strain-figure  is  merely  transferred 
from  one  part  of  that  body  to  another  part.  (The  vortex- 
theory,  you  will  remember,  requires  us  to  suppose  that  an 
atom  always  consists  of  the  same  portion  of  ether ;  and  in 
this  respect  it  is  diametrically  opposed  to  Dr.  Burton's 
theory.)  Gravitation  and  other  inter-moleQular  and  inter- 
atomic forces  are  assumed  to  arise  from  the  distribution  of 
stress  that  accompanies  the  strains  in  the  strain-figures. 
Other  consequences  of  the  strain-figure  hypothesis  have  been 
examined,  but  the  hypothesis  is  so  new  and  so  imperfectly 
developed  that  it  will  hardly  be  profitable  to  discuss  it  further 
in  this  place.  It  is  extremely  ingenious  and  interesting,  but 
we  must  wait  for  further  researches  before  we  can  pronounce 
upon  its  adequacy  or  inadequacy. 

Internal  Vibration  of  Molecules.  —  The  various  phenomena 
of  physics  and  astronomy  compel  us  to  admit  that  matter  can 
move  through  the  ether  freely,  without  experiencing  the  least 
resistance.  But  since  we  know  that  matter  can  emit  radiant 
energy,  it  follows  that  there  are  modes  of  molecular  motion 
that  can  be  communicated  to  the  ether  ;  and  we  are  impelled 
to  the  belief  that  it  is  the  internal  vibratory  energy  that  is 
transmitted  in  this  way.  That  such  energy  exists,  is  quite 
evident ;  for  if  the  molecules  of  bodies  are  elastic,  we  must 
suppose  them  to  be  capable  of  some  sort  of  internal  vibration. 
Let  us  consider  a  gas,  assuming  that  its  molecules  have 
definite  masses  and  definite  sizes,  and  that  for  each  of  them 
there  is  a  certain  shape  in  which  the  internal  stresses  are 
either  zero,  or  at  least  a  minimum.  When  two  such  mole- 


INTERNAL    VIBRATION    OF    MOLECULES.  183 

cules  collide,  we  must  suppose  that  the  collision  throws  each 
of  them  into  violent  vibration,  just  as  a  stretched  string  is 
thrown  into  vibration  upon  being  struck  with  a  hammer.  We 
know  that  the  string  can  vibrate  in  different  ways  :  it  may 
vibrate  as  a  whole,  or  in  two  equal  segments  separated  by  a 
node,  or  in  three  such  segments,  or  four,  or,  in  general,  in  any 
number  of  them.  When  such  a  string  is  struck  we  usually 
find  that  all  of  these  possible  modes  of  vibration  occur  simul- 
taneously, so  that  the  actual  motion  is  very  complex.  In  the 
stretched  string  the  periodic  times  of  the  various  possible 
vibrations  are  proportional  to  the  roots  of  the  equation 

sin  |-  1  =  0; 


that  is,  they  are  inversely  proportional  to  the  simple  numbers 
1,  2,  3,  4,  ...  Doubtless  there  is  some  similar  law  connecting 
the  periodic  times  of  the  possible  vibrations  that  may  occur 
in  elastic  molecules  ;  though  we  cannot  suppose  that  law  to 
be  as  simple  as  the  one  connecting  the  various  periodic  times 
of  the  string.  In  discussing  this  question  of  the  vibration- 
periods  of  elastic  systems,  Professor  J.  J.  Thomson  tells  us 
that  "if  the  vibrating  system  .  .  .  were  like  a  bar,  the 
periods  would  be  proportional  to  the  natural  numbers  for  the 
longitudinal  and  torsional  vibrations,"  and  to  the  reciprocals 
of  the  roots  of  the  equation 


for  the  transverse  vibrations.  "  If  the  system  were  a  circular 
membrane/7  he  continues,  "the  frequencies  would  be  pro- 
portional to  the  roots  of  an  equation  formed  by  equating  a 
Bessel's  function  to  zero.  If  the  system  were  a  uniform 
elastic  sphere,  the  frequencies  would  be  the  roots  of  a  compli- 
cated equation  given  by  Chree  in  the  Transactions  of  the 
Cambridge  Philosophical  Society.  Other  periods  which  have 
been  worked  out  are  those  of  circular  vortex  rings.  The 


184      THE  MOLECULAK  THEORY  OF  MATTER. 

frequencies  of  the  higher  vibrations  [of  such  rings]  about  the 
circular  form  are  proportional  to 


where  n  is  a  large  natural  number  [i.e.  integer],  and  the 
vibrations  about  the  circular  cross-section  are  proportional  to 
the  natural  numbers."*  We  have  to  think  of  a  gas-molecule 
as  colliding  with  another  one,  and  then  flying  off  through  the 
ether  in  a  sensibly  straight  line  until  it  again  experiences  a 
collision.  At  each  collision  vibrations  are  set  up  within  the 
molecule,  and  in  the  interval  between  successive  collisions  — 
that  is,  while  the  molecule  is  describing  its  "  free  path  "  —  it 
vibrates  according  to  its  own  proper  periods,  and  the  ether  in 
which  it  is  submerged  picks  up  these  vibrations  and  carries 
them  away  as  radiant  heat,  or  as  light.  You  must  not 
suppose,  however,  that  the  process  is  exactly  analogous  to 
what  occurs  when  a  particle  immersed  in  a  jelly  is  caused  to 
vibrate.  This  was  indeed  believed  to  be  the  case  when  the 
elastic-solid  theory  of  light  was  held  to  be  true,  but  when 
that  theory  was  discarded  it  became  evident  that  the  real 
phenomena  are  essentially  different  from  those  suggested  by 
the  jelly-analogue ;  and  I  think  we  are  still  a  long  way  from 
knowing  precisely  what  does  take  place  when  a  vibrating 
molecule  gives  up  its  energy  to  the  ether.  You  will  observe 
that  writers  on  the  theory  of  light  merely  consider  what 
occurs  in  the  ether  as  it  transmits  the  light,  and  do  not 
attempt  to  trace  the  exact  processes  by  which  the  ethereal 
motions  originate.  There  can  be  no  doubt  that  here  is  a 
fruitful  field  for  investigation,  but  at  present  we  are  hardly 
prepared  to  enter  upon  it.  Thus  far  I  have  referred  only  to 
the  vibrations  of  #as-molecules,  and  I  have  said  that  while  a 
molecule  is  traversing  its  free  path,  the  vibrations  that  occur 
are  performed  in  accordance  with  its  natural  vibration-periods. 
At  the  instant  of  collision,  and  for  an  extremely  short  time 

*  Watts's  Dictionary  of  Chemistry  (new  edition),   article  Molecular 
Constitution  of  Bodies. 


INTERNAL    VIBRATION   OF   MOLECULES.  185 

afterwards,  the  vibrations  probably  differ  more  or  less  from 
these  periods,  on  account  of  the  extreme  violence  of  the  intra- 
molecular shocks.  Careful  attention  should  be  paid  to  this 
point,  since  it  shows  that  in  solids  or  liquids,  where  there  is 
practically  no  free  path,  and  in  highly  compressed  gases  where 
the  free  path  is  very  short,  we  cannot  expect  to  find  the  com- 
parative simplicity  of  vibration  that  we  do  find  in  gases  under 
ordinary  conditions  of  density.  The  data  for  investigating 
the  vibration-periods  of  molecules  are  furnished  by  that 
simple  yet  ingenious  and  powerful  instrument  known  as  the 
spectroscope,  which  enables  us  to  analyze  the  complicated 
vibratory  motion  that  they  have  impressed  upon  the  ether, 
and  to  examine  separately  the  constituent  simple  vibrations 
of  which  it  is  composed.  Professor  E.  C.  Kedzie  says  of  this 
wonderful  instrument,  "  If  there  was  ever  a  flank  movement 
on  nature  by  which  she  has  been  compelled  to  surrender  a 
part  of  her  secrets  it  was  the  discovery  of  the  spectroscope, 
'  which  enables  us  to  peer  into  the  very  heart  of  nature ' "  ;  * 
and  Maxwell  says,  though  I  think  his  statement  is  far  too 
strong,  "  An  intelligent  student,  armed  with  the  calculus  and 
the  spectroscope,  can  hardly  fail  to  discover  some  important 
fact  about  the  internal  constitution  of  a  molecule."  f  By  the 
aid  of  this  instrument  the  vibration-periods  of  molecules  have 
been  patiently  studied  and  tabulated,  $  and  many  attempts 
have  been  made  to  find  some  relation  among  them,  analogous 
to  the  integer-law  that  holds  for  stretched  strings,  and  to  the 
other  laws  that  I  have  told  you  about  in  connection  with  bars 
and  membranes  and  spheres.  No  very  great  success  has 
rewarded  these  efforts,  yet  something  has  been  done,  and 
more  is  sure  to  follow.  Hydrogen,  on  account  of  its  chemical 
and  physical  properties,  has  long  been  regarded  as  a  compara- 
tively simple  substance ;  and  especial  attention  has  been  paid 

*  Proceedings  of  the  American  Association  for  the  Advancement  of 
Science,  August  meeting,  1891,  page  162. 

t  Nature,  March  11,  1875. 

I  See,  for  example,  Watts's  Index  of  Spectra  (Manchester,  England, 
Abel  Hey  wood  &  Son,  1889). 


186 


THE  MOLECULAR  THEORY  OF  MATTER. 


to  it  on  that  account,  in  the  hope  that  the  relation  among  its 
various  vibration-periods  might  prove  to  be  comparatively 
simple,  and  might  therefore  be  the  more  readily  found. 
Experience  has  shown  that  this  hope  was  justifiable ;  for  a 
remarkable  relation  among  the  vibration-periods  of  hydrogen 
has  been  discovered  by  Balmer.  The  relation  in  question  is 
this  :  If  the  different  lines  in  the  spark-spectrum  of  hydrogen 
be  numbered  consecutively,  calling  the  Ha.  line  3,  the  next  one 
4,  and  so  on,  then  the  wave-length  of  the  line  whose,  number 
is  m  is  _2 


X  =  3645.42 


—  4* 


(60) 


In  the  following  table  the  results  of  this  formula  are  com- 
pared with  the  observed  facts. 

SPARK-SPECTRUM  OF  HYDROGEN. — BALMER' s  LAW. 


LUTE. 

m. 

WAVE-LENGTH. 

DIFFERENCE. 

CALCULATED. 

OBSERVED. 

Ha 

3 

6562.8 

6563.1 

+  .3 

H{3 

4 

4860.6 

4860.7 

+  .1 

Hy 

5 

4339.8 

4339.5 

—.3 

H3 

6 

4101.1 

4101.2 

+.1 

He 

7 

3969.5 

3969.2 

-.3 

H{ 

8 

3888.4 

3888.1 

-.3 

H-n 

9 

3834.8 

3834.9 

+.1 

He 

10 

3797.3 

3797.3 

.0 

Hi 

11 

3770.0 

3769.9 

-.1 

HK 

12 

3749.6 

3750.2 

+  .6 

H\ 

13 

3733.8 

3734.1 

+  .3 

Hn 

14 

3721.4 

3721.1 

-.3 

Hv 

15 

3711.4 

3711.2 

-.2 

*  The  vibration-period  is  proportional  to  the  wave-length  —  being 
equal,  in  fact,  to  the  wave-length  divided  by  the  velocity  of  light. 
Balmer 's  law  can  easily  be  written  so  as  to  give  the  vibration-periods 
directly  ;  but  most  writers  state  it  in  connection  with  the  wave-lengths, 
and  I  have  thought  best  to  follow  the  custom  thus  established. 


INTERNAL   VIBRATION    OF   MOLECULES.  187 

The  agreement  between  the  calculated  and  observed  wave- 
lengths is  very  good ;  and  we  may  conclude  that  the  hydrogen 
molecule  is  so  constituted  that  equation  (60)  represents  all 
the  various  kinds  of  elastic  vibration  that  are  possible  within 
it,  under  the  conditions  that  prevail  when  the  spark-spectrum 
of  the  gas  is  being  examined.  Further  than  this  we  cannot 
go,  at  present,  because  no  one  has  shown  what  sort  of  a  body 
would  have  the  series  of  vibration-periods  that  is  represented 
by  (60).  I  have  compared  the  light-producing  vibrations  of 
a  molecule  to  the  sound-producing  vibrations  of  a  sonorous 
body ;  but  I  must  caution  you  against  supposing  that  there  is 
any  very  close  analogy  between  the  two.  I  have  referred  to 
the  familiar  phenomena  of  sound,  in  order  to  assist  your 
imaginations  a  little  ;  but  you  should  understand  clearly  that 
the  ultimate  phenomena  of  light  are  probably  quite  different. 
Since  the  electrical  nature  of  light  has  been  recognized,  the 
suggestion  has  been  made  that  molecules  behave  like  con- 
ductors in  which  oscillatory  electrical  discharges  take  place, 
the  form,  capacity,  and  resistance  of  the  molecules  determin- 
ing the  rapidity  of  the  discharges,  and  hence  also  the 
positions  of  the  spectral  lines.  I  do  not  think  this  conception 
adds  much  to  our  knowledge  of  the  molecule,  since  I  can 
think  of  an  "  electrical  discharge  "  only  as  a  motion  of  some 
kind,  in  which  the  molecule  and  the  ether  probably  both 
participate;  nor  do  I  think  that  any  other  hypothesis  is 
likely  to  help  us  much  until  we  have  a  more  exact  knowledge 
of  the  kind  of  motion  that  occurs  in  the  free  ether  when  a 
ray  of  light  is  passing  through  it.  Many  physicists  appear 
to  regard  molecules  as  aggregates  of  smaller  particles,  which 
are  held  together  by  a  system  of  attractive  forces,  and  which 
execute  to-and-fro  oscillations  of  definite  periods  when  the 
equilibrium  of  the  system  is  disturbed  by  a  collision,  or  in 
any  other  manner ;  the  light-waves  being  supposed  to  arise 
from  these  oscillations.  Thus  Professor  J.  J.  Thomson  says, 
that  Balmer's  law  seems  to  show  "  that  the  hydrogen  molecule; 
is  a  system  possessing  an  infinite  number  of  degrees  of  free- 


188      THE  MOLECULAR  THEORY  OF  MATTER. 

dom,  and  not  a  finite  number  of  rigid  particles  mutually 
attracting  each  other."  So  far  as  the  to-and-fro  theory  of 
the  origin  of  light  is  concerned,  I  think  we  may  dismiss  it 
altogether ;  for  we  have  already  assumed  that  the  ether  does 
not  absorb  energy  from  a  particle  moving  through  it,  and  it 
is  therefore  difficult  to  conceive  how  light  can  be  produced  by 
any  combination  of  such  to-and-fro  motions.  Professor  Thom- 
son would  probably  admit  this  point  readily  enough,  and  I  am 
inclined  to  think  that  his  remark  has  a  deeper  significance  — 
that  it  is  aimed,  in  fact,  at  Boltzmann's  law  of  the  partition 
of  kinetic  energy  among  the  different  degrees  of  freedom  of 
gas-molecules ;  for  if  that  law  be  true,  it  follows  that  the 
number  of  degrees  of  freedom  of  a  molecule  is  finite.*  For 
some  reason  or  other  Boltzmann's  theorem  has  given  rise  to  a 
good  deal  of  controversy,  and  English  mathematicians,  as  a 
rule,  appear  to  be  distinctly  hostile  to  it,  although  I  cannot 
quite  see  why.  As  I  understand  the  theorem,  it  relates  only 
to  the  kinetic  energy  of  translation  and  rotation  of  the  parts 
of  a  molecule,  and  not  to  those  modes  of  vibration  which  are 
analogous  to  the  motions  of  a  sounding  wire  or  bell  or  other 
elastic  body.  The  general  facts  that  have  been  ascertained 
about  the  molecules  of  gases  appear  to  be  these  :  The  kinetic 
energy  of  the  molecules  is  divisible  into  two  parts,  one  of 
which  is  enormously  greater  than  the  other.  The  greater 
part  consists  in  various  motions  of  translation  and  rotation, 
either  of  the  molecule  as  a  whole  or  of  its  parts  among  them- 
selves ;  and  it  is  this  portion  of  the  kinetic  energy  which  is 
divided  up  equally  among  the  different  "  degrees  of  freedom  " 
of  the  molecules  (using  the  phrase  in  its  restricted  sense). 
The  remaining  small  part  of  the  total  kinetic  energy  consists 
in  elastic  vibratory  motion,  within  the  very  substance  of  the 
molecule  or  its  parts ;  and  to  this  almost  infinitesimal  portion 
of  the  kinetic  energy  Boltzmann's  law  does  not  apply.  The 
motions  of  translation  and  rotation  are  not  impeded  by  the 

*  See  the  section  on  "Generalized  Theorems"  (page  34),  and  that  on 
.the  "Ratio  of  the  Specific  Heats  of  Gases"  (page  47). 


INTEKNAL   VIBRATION   OF   MOLECULES.  189 

ether  ;  but  the  internal  vibratory  motion  is  of  such  a  character 
that  the  ether  absorbs  it,  and  bears  it  away  as  radiant  energy. 
If  a  gas  receives  no  heat  from  without,  it  will  be  cooled  by 
the  continual  abstraction  of  vibratory  energy  until  its  tempera- 
ture falls  to  the  absolute  zero.  It  is  true  that  only  a  small 
part  of  the  total  kinetic  energy  exists  at  any  one  time  in  the 
form  of  energy  of  vibration,  but  it  is  also  true  that  the  vibra- 
tory energy  can  never  entirely  disappear  while  the  gas  con- 
tains any  kinetic  energy  whatever ;  for  some  vibratory  energy 
must  exist,  so  long  as  there  are  molecular  collisions.  As  the 
vibratory  energy  is  removed  by  the  ether,  more  will  be  pro- 
duced, at  the  expense  of  energy  of  other  kinds  ;  and  the  end 
of  the  process  will  come  only  when  the  molecules  have  all 
come  to  rest.  If  the  gas  is  exposed  to  radiations  emanating 
from  an  external  heat-source,  the  phenomena  are  reversed  — 
the  vibratory  energy  of  the  molecules  increases  by  absorption 
of  the  ethereal  vibrations,  and  the  energy  so  gained  becomes 
presently  transformed  into  energy  of  other  kinds,  by  means 
of  the  collisions  that  are  constantly  occurring  among  the 
molecules  ;  and  when  the  kinetic  energy  of  translation  of  the 
molecules  has  become  sensibly  increased  by  this  process,  we 
say  that  the  gas  has  grown  "  warmer.7'  The  well-known 
difficulty  of  heating  gases  by  simple  radiation  shows,  how- 
ever, that  the  molecules  of  these  bodies  do  not  readily  absorb 
vibratory  energy  from  the  ether.  I  do  not  know  why  this  is 
so  :  solid  bodies  take  up  such  energy  readily  enough,  and  it  is 
not  easy  to  account  for  the  diathermacy  of  gases.  Before 
leaving  this  subject  I  must  say  that  equation  (60)  is  by  no 
means  general.  It  applies  only  to  hydrogen,  and  we  cannot, 
by  changing  the  constants,  make  it  represent  the  vibration- 
periods  of  other  bodies.  Numerous  investigators  have  sought 
for  similar  relations,  however,  that  should  hold  true  for  other 
elements,  and  with  some  slight  degree  of  success.  Thus 
Kayser  and  Kunge  found  that  the  various  wave-lengths  of 
lithium,  sodium,  potassium,  rubidium,  and  caesium  are  expres- 
sible by  equations  of  the  form 


190     THE  MOLECULAR  THEORY  OF  MATTER. 


. 

m       m 

Numerous  formulae  of  this  sort  have  in  fact  been  proposed, 
but,  so  far  as  I  am  aware,  none  of  them  is  as  satisfactory  as 
Balmer's.  The  spectroscope  offers  us  a  vast  fund  of  informa- 
tion, and  yet  for  theoretical  purposes  it  is  nearly  useless, 
because  the  key  by  which  the  riddle  is  to  be  read  still  remains 
unfound. 

Gravitation.  —  It  is  strange  that  no  satisfactory  theory  of 
the  nature  of  gravitation  has  yet  been  proposed;  for  the 
phenomena  to  be  explained  are  simple,  and  they  are  familiar 
to  everybody.  The  general  fact  of  gravitation  is,  that  between 
every  pair  of  material  particles  there  exists  an  attractive 
force  which  is  proportional  to  the  product  of  the  masses  of 
the  two  particles,  and  to  the  reciprocal  of  the  square  of  the 
distance  between  them.  Various  attempts  to  account  for  this 
attraction  have  been  made,  and  although  all  the  theories  yet 
proposed  have  been  eminently  unsatisfactory,  it  may  be  of 
interest  to  review  a  few  of  them  briefly.  Newton  naturally 
gave  some  attention  to  the  problem,  and  suggested  that  the 
ether  is  everywhere  in  a  state  of  pressure,  but  that  for  some 
reason  or  other  this  pressure  is  less  in  the  neighborhood  of 
dense  bodies  than  it  is  elsewhere.  He  showed  that  such  a 
state  of  things  would  cause  two  bodies  immersed  in  the  ether 
to  be  urged  towards  each  other,  and  he  also  showed  that  if 
the  diminution  of  pressure  at  any  point,  due  to  the  presence 
of  the  dense  body,  were  inversely  proportional  to  the  distance 
of  that  point  from  the  body,  the  apparent  attraction  would 
obey  the  law  of  inverse  squares.  He  was  unable,  however,  to 
imagine  any  physical  cause  for  such  a  distribution  of  ether- 
pressure.  The  most  famous  theory  of  gravitation  is  undoubt- 
edly that  of  the  Swiss  philosopher  Le  Sage.  According  to 
him  there  is  an  enormous  number  of  extremely  small,  "  ultra- 
mundane "  corpuscles  of  some  sort  or  other,  flying  about 
through  space  with  tremendous  speed,  and  in  every  conceiv- 


GRAVITATION.  191 

able  direction.  If  there  were  only  one  body  in  space  it  would 
be  bombarded  equally  on  all  sides  by  the  corpuscles,  and  the 
impact-forces  exerted  upon  it  would  be  in  equilibrium.  But 
Le  Sage  conceived  that  if  there  were  two  bodies  in  space,  each 
of  them  would  shield  the  other  one  to  a  certain  extent,  so 
that  the  bombardment  that  either  body  received  would  be 
most  severe  on  the  side  remote  from  the  other  one,  and  hence 
the  two  bodies  would  be  urged  together,  and  there  would  be 
an  apparent  attraction  between  them.  There  are  a  host  of 
objections  to  this  theory,  and  we  cannot  even  momentarily 
consider  it  to  be  true.  For  example,  in  order  to  explain  why 
a  great  amount  of  heat  is  not  produced  by  the  collisions,  we 
have  to  assume  that  the  corpuscles  rebound  with  undiminished 
energy.  But  in  that  case,  although  the  bodies  would  still 
shield  each  other,  as  before,  from  the  impact  of  corpuscles 
coming  directly  from  the  depths  of  space,  we  must  note  that 
each  body  would  also  reflect  corpuscles  in  all  directions,  so 
that  its  neighbor  would  be  struck  by  a  certain  number  of 
corpuscles  that  would  otherwise  have  missed  it ;  and  it  has 
been  shown  that  the  impact  against  one  body  of  the  corpus- 
cles reflected  from  the  other  one  would  just  suffice  to  annul 
the  effect  of  the  direct  shielding  action,  and  to  prevent  the 
realization  of  any  gravitative  tendency  whatever.  Le  Sage's 
original  theory  can  be  modified  so  as  to  avoid  this  difficulty, 
but  it  will  hardly  be  profitable  for  us  to  discuss  such  modifi- 
cations, because  there  are  so  many  other  objections  to  the 
theory  that  it  now  has  only  a  historic  interest.  For  example, 
in  order  to  account  for  the  fact  that  gravitative  action  varies 
as  the  mass  of  a  body,  and  not  as  its  surface,  we  have  to  sup- 
pose that  the  "  ultramundane  corpuscles  "  can  pass  through 
ordinary  matter  quite  freely,  so  as  to  strike  all  its  molecules 
with  substantially  the  same  frequency ;  and  this  implies  a 
more  open  structure  of  matter  than  we  can  readily  reconcile 
with  what  we  know  of  the  sizes  of  molecules,  and  of  inter- 
molecular  distances.  The  "ether-squirt"  theory  of  gravita- 
tion assumes  that  each  particle  of  matter  is  a  sort  of  center  at 


192     THE  MOLECULAR  THEORY  OF  MATTER. 

which  ether  is  continuously  created,  so  that  from  every  such 
particle  a  ceaseless  stream  of  ether  flows  out  in  all  directions. 
Such  a  state  of  things  would  give  rise  to  an  apparent  attrac- 
tion, similar  to  gravitation ;  but  as  the  theory  requires  us  to 
admit  a  wholesale  and  perpetual  production  of  ether  from 
nothing,  we  must  conclude  that  the  probability  of  its  truth  is 
no  greater  than  the  euphony  of  its  name.  Similar  remarks 
apply  to  the  "  ether-sink  "  theory,  which  differs  from  the  one 
we  have  just  considered  only  in  assuming  that  at  each  mole- 
cule there  is  a  destruction  of  ether,  instead  of  a  creation  of  it. 
The  so-called  " vibratory  theory"  of  gravitation  is  more 
interesting  than  any  that  we  have  yet  considered.  You  will 
remember  that  I  said  that  the  ether  is  believed  to  be  incapable 
of  transmitting  waves  of  compression  and  rarefaction  (analo- 
gous to  sound-waves),  because  no  phenomena  could  be  dis- 
covered which  could  be  attributed  to  such  waves.  It  has 
been  suggested  that  gravitation  may  transpire  to  be  the 
missing  phenomenon,  and  that  the  attraction  between  two 
bodies  may  be  due  to  the  mutual  action  of  the  compression- 
waves  and  rarefaction-waves  that  emanate  from  them.  Con- 
siderable attention  has  been  paid  to  this  hypothesis,  both 
mathematically  and  experimentally.  It  has  been  shown  that 
a  tuning-fork  vibrating  in  the  air  can  attract  a  light  pith  ball, 
and  other  phenomena  of  a  like  nature  have  been  observed. 
The  mathematical  investigations  that  the  theory  calls  for  are 
exceedingly  difficult,  and  I  am  not  aware  that  they  have 
yielded  any  conclusive  results.  It  appears  to  be  true,  how- 
ever, that  a  particle  would  be  attracted  toward  the  center  of 
disturbance  if  its  density  were  greater  than  that  of  the  sur- 
rounding medium,  and  its  dimensions  small  in  comparison 
with  the  length  of  the  waves.  This  result,  so  far  as  it  goes, 
is  favorable  to  the  wave-theory ;  for  matter  is  universally 
admitted  to  be  denser  than  ether,  and  there  is  good  reason  for 
believing  that  if  gravitation-waves  exist  at  all,  they  are  very 
long.  In  connection  with  this  theory  Lord  Kelvin  has  called 
attention  to  "  the  general  principle  that  in  fluid  motion  the 


GRAVITATION.  193 

average  pressure  is  least  where  the  average  energy  of  motion 
is  greatest."  Now  if  the  wave-theory  of  gravitation  is  true, 
the  vibratory  energy  of  the  ether  would  be  greatest  in  the 
immediate  neighborhood  of  molecules  of  matter ;  and  hence 
the  ether-pressure  would  be  least  at  such  places,  and  we 
should  have  a  distribution  of  pressure  something  like  that 
demanded  by  Newton's  theory  of  gravitation.  The  wave- 
theory  is  quite  interesting,  but  unfortunately  there  are  certain 
grave  objections  to  it,  some  of  which  appear  to  be  absolutely 
fatal.  In  the  first  place  the  theory  requires  us  to  admit  that 
a  body  exerts  a  greater  attractive  power  when  hot  than  it 
does  when  cold ;  because  when  the  body  is  hot  its  molecules 
are  vibrating  more  energetically,  and  hence  the  waves  that 
they  emit  have  a  greater  amplitude.  No  such  phenomenon 
has  ever  been  detected.  Again,  if  gravitative  attraction  is 
due  to  a  wave-motion  in  the  ether,  we  should  have  to  admit 
that  it  has  a  finite  speed  of  propagation.  Nothing  of  the 
kind  has  been  observed ;  but  we  know  from  astronomical  con- 
siderations that  if  there  is  any  such  a  finite  speed  of  propa- 
gation, it  is  certainly  greater  than  a  million  times  the  speed 
of  light.  This  naturally  suggests  that  gravitation  is  not  due 
to  any  sort  of  wave-motion  in  the  medium  that  transmits 
light.  In  Maxwell's  theory  of  gravitation  it  is  assumed  that 
bodies  produce  a  stress  in  the  ether  about  them,  of  such  a 
nature  that  there  is  a  pressure  along  the  lines  of  gravitative 
force,  combined  with  an  equal  tension  in  all  directions  at  right 
angles  to  those  lines.  (In  the  case  of  a  single  body  in  space, 
the  pressures  would  be  radial,  and  the  surfaces  of  equal 
tension  would  be  concentric  spheres  described  about  the  body 
as  a  center ;  the  stresses  in  the  ether  about  the  body  being 
somewhat  similar  to  those  that  exist  in  a  cannon  at  the 
moment  of  discharge.)  "  Such  a  state  of  stress,'7  says  Max- 
well, "would  no  doubt  account  for  the  observed  effects  of 
gravitation.  We  have  not,  however,  been  able  hitherto  to 
imagine  any  physical  cause  for  such  a  state  of  stress.  It  is 
easy  to  calculate  the  amount  of  this  stress  which  would  be 


194      THE  MOLECULAR  THEORY  OF  MATTER. 

required  to  account  for  the  actual  effects  of  gravity  at  the 
surface  of  the  earth.  It  would  require  a  pressure  of  37,000 
tons'  weight  on  the  square  inch  in  a  vertical  direction,  com- 
bined with  a  tension  of  the  same  numerical  value  in  all  hori- 
zontal directions.  The  state  of  stress,  therefore,  which  we 
must  suppose  to  exist  in  the  invisible  medium,  is  3,000 
[1,000]  times  greater  than  that  which  the  strongest  steel 
could  support."  *  Maxwell's  theory  is  somewhat  promising, 
but  I  think  we  cannot  say  more  than  this  of  it  until  a  sufficient 
cause  for  the  ether-stresses  can  be  found.  We  may  now  pass 
to  the  consideration  of  Lord  Kelvin's  vortex-theory  of  the 
constitution  of  molecules ;  and  here  we  find  that  there  is  some 
small  hope  of  explaining  gravitation,  though  it  is  of  a  purely 
negative  character.  It  is  known  that  vortices  exert  a  sensible 
influence  upon  one  another,  even  when  they  are  a  considerable 
distance  apart,  and  for  certain  special  cases  this  influence  has 
been  investigated  with  a  considerable  approach  to  precision ; 
but  the  vortex-theory  has  not  yet  been  developed  sufficiently 
to  enable  us  to  investigate  the  interaction  of  vortices  with 
absolute  precision,  and  there  are,  undoubtedly,  certain  residual 
effects  which  are  not  included  in  the  approximate  equations 
that  form  the  basis  of  what  we  now  know  about  vortices. t  It 
is  possible  that  these  neglected  residual  effects  will  prove  to 
be  sufficient  to  account  for  gravitative  attraction ;  for  gravi- 
tation, as  is  well  known,  is  an  extremely  weak  force,  becoming 
sensible  only  when  bodies  of  enormous  size  are  involved. 
The  difficulty  of  deciding  this  point  is  tremendous,  and  at 
present  we  can  only  say  that  the  vortex-theory  may  possibly 
be  competent  to  explain  gravitation.  Professor  Thomas 
Preston  makes  an  interesting  suggestion  about  gravitation, 
which  I  will  quote  to  you,  although  I  do  not  think  it  is  put 
forward  as  being  in  any  degree  probable.  I  have  told  you 

*  Encyclopaedia  Britannica,  article  Attraction. 

t  For  a  mathematical  discussion  of  the  interaction  of  vortices,  see 
Professor  J.  J.  Thomson's  Treatise  on  the  Motion  of  Vortex  Eings 
(London,  Macmillan  &  Co.,  1883). 


GRAVITATION.  195 

that  a  finite  ether-vortex  must  either  return  into  itself  so  as 
to  form  a  closed  curve,  or  must  have  its  ends  against  a  bound- 
ing surface  of  the  ether.  Thus  it  would  be  possible  to  have 
vortices  that  do  not  return  into  themselves,  but  which  have 
ends  that  abut  against  molecules  of  dense  matter.  With  this 
possibility  in  mind  Professor  Preston  says,  "We  might  sup- 
pose a  body  connected  to  the  earth  by  vortex  filaments  in  the 
ether,  which  would  replace  the  lines  of  force.  The  ether  is 
spinning  round  these  lines,  and  when  the  body  is  lifted  from 
the  earth  the  work  done  is  expended  in  increasing  the  length 
of  the  vortex  filaments.  The  work  is  thus  being  stored  up  as 
energy  of  motion  of  the  ether,  and  when  the  body  falls  to 
earth  the  vortex  lines  diminish  in  length,  and  their  energy  of 
motion  passes  into  the  body  and  is  represented  by  the  kinetic 
energy  of  the  mass.'7*  Professor  Preston  probably  intended 
this  suggestion  as  a  mere  illustration  of  the  possibility  of 
explaining  gravitation ;  for  it  would  be  quite  extravagant  to 
imagine  every  molecule  in  the  universe  to  be  united  to  every 
other  one  by  a  vortex  filament  —  space  could  hardly  contain 
such  a  tangle.  Moreover,  since  vortices  cannot  intersect,  a 
few  seconds  of  intermolecular  motion  would  suffice  to  tie  up 
the  vortex-system  into  a  mass  of  knots  that  would  drive 
Gordius  mad  with  envy,  and  render  the  "  first  law  of  motion  " 
impossible.  Dr.  Burton,  in  connection  with  his  strain-figure 
theory  of  the  constitution  of  molecules,  has  suggested  that 
molecules  do  not  have  definite  sizes,  but  that  in  the  ether 
surrounding  each  molecular  center  there  are  stresses  and 
strains  which  grow  continually  less  as  we  pass  away  from 
that  center,  but  which  never  absolutely  vanish.  In  this  case 
every  molecule  in  the  universe  would  exert  some  influence  on 
every  other  molecule.  Dr.  Burton  shows  that  in  its  most 
general  form  his  theory  could  account  for  either  attraction  or 
repulsion,  but  that  by  making  a  certain  very  simple  assump- 
tion about  the  nature  of  the  strain-figures,  the  theory  could  be 

*  Thomas  Preston,   Theory  of  Heat  (London  and  New  York,  Mac- 
millan  &  Co.,  1894),  page  90. 


196      THE  MOLECULAR  THEORY  OF  MATTER. 

modified  so  that  the  forces  acting  between  distant  molecules 
should  always  be  attractive ;  yet  even  if  the  strain-figure 
theory  should  afford  a  perfectly  satisfactory  explanation  of 
gravitation,  we  could  not  logically  accept  that  explanation 
until  the  fundamental  conceptions  of  the  original  hypothesis 
were  shown  to  be  in  harmony  with  all  the  other  phenomena 
of  matter.  I  have  now  reviewed,  briefly,  some  of  the  more 
famous  and  interesting  theories  that  have  been  proposed  to 
account  for  gravitative  attraction,  and  it  is  easy  to  see  that 
they  all  fail  in  some  important  particular.  It  may  be  that 
the  future  will  bridge  over  some  of  these  failures.  It  may  be, 
on  the  other  hand,  that  we  have  been  looking  at  the  problem 
from  the  wrong  point  of  view,  altogether. 


Conclusion.  —  We  have  examined  the  molecular  theory  of 
matter  as  it  stands  to-day,  and  we  have  found  that  something 
is  known  of  the  constitution  of  gases,  a  little  about  the 
constitution  of  liquids,  and  almost  nothing,  for  certain,  of 
solids.  So  far  as  the  sizes  of  molecules  are  concerned,  we 
have  seen  that  it  is  possible  to  discover  the  general  order  of 
their  magnitude ;  and  we  have  also  seen  that  nothing  what- 
ever is  known  about  their  constitution,  or  about  the  mechani- 
cal nature  of  intermolecular  forces.  We  are  still  very  far 
from  having  a  complete  theory  of  matter  —  our  knowledge  of 
it  is  in  fact  very  fragmentary  —  and  yet  there  is  strong  reason 
to  believe  that  we  are  working  in  the  right  direction.  A  great 
deal  has  been  done  in  the  last  forty  years,  and  it  is  just 
possible  that  within  the  next  forty  the  world  will  be  fortu- 
nate enough  to  produce  a  great  genius  who  shall  coordinate 
the  isolated  facts  that  we  now  have,  fill  up  the  vast  gaps  in 
our  present  knowledge,  and  provide  us  with  a  classical  treatise 
on  The  Constitution  of  Matter,  which  shall  be  worthy  to  stand 
on  our  shelves  beside  the  immortal  Principia  of  Sir  Isaac 
Newton. 


APPENDIX. 


On  the  Integration  of  Certain  Equations  in  the  Text.  —  In 

deriving  equations  (2)  and  (4),  and  in  constructing  the  table 
on  page  26,  it  is  necessary  to  perform  certain  integrations  that 
will  doubtless  prove  troublesome  to  the  student  unless  he  has 
devoted  more  time  to  the  integral  calculus  than  is  usual  in 
our  colleges  and  scientific  institutions.  The  processes  by 
which  these  integrations  may  be  effected  will  therefore  be 
briefly  indicated.  It  is  easily  shown,  by  direct  differentiation, 
that 

(L  (x        '  c      )  —  ( fi      J. )  X          €       dx      Zi  e         X  dx. 

If  we  integrate  this  equation,  term  by  term,  and  then  trans- 
pose and  divide  by  2,  we  have  the  formula 

/V*2  •  xndx  =  —  4-cc"-1  -  c-*2  +  ?t— i  fxn~2  •  <rx*dx.   (61) 

By  successive  applications  of  this  formula  any  integral  having 
the  form  of  the  left-hand  member  of  (61)  can  be  made  to 
depend  upon  another  integral  of  the  same  form,  but  with  the 
exponent  of  x  unity  (if  n  is  odd)  or  zero  (if  n  is  even).  Hence 
every  integral  of  this  form  depends,  ultimately,  on  one  or 
other  of  the  two  following  forms  : 

/  e"*2  xdx,  or     /  e"*2  dx.  (62) 

The  first  of  these  is  immediately  integrable ;  for  if  we  multiply 
it  by  —  2  it  becomes 

/-  2  2  -x* 

* 


198  APPENDIX. 

The  second  one  is  much  more  difficult  to  handle,  and  we  may 
distinguish  two  cases  —  (1)  when  the  limits  of  the  integration 
are  both  either  infinite  or  zero,  and  (2)  when  one  (or  both)  of 
them  is  a  finite  quantity  other  than  zero.  If  the  limits  are 
0  and  oo  the  integration  may  be  effected  by  the  following 
artifice.  Consider  the  surface  whose  equation  is 

«  =  €-(**  +  »•>.  (63) 

The  volume  included  between  this  surface  and  the  plane 
«  =  0,  and  between  the  x-limits  0  and  oo  ,  and  the  y-limits 
0  and  oo  ,  is 

V  =CC<r^  +  "°>  dx  dy.  (64) 

0        0 

If  we  represent  the  integral 


(65) 

by  u,  then,  integrating  (64)  with  respect  to  #,  we  have 

*dy.  (66) 


Now  the  integral  expressed  in  (66)  is  evidently  the  same  as 
(65),  since  its  value  cannot  depend  upon  the  particular  symbol 
that  we  use  to  represent  the  variable  quantity.  But  we  have 
represented  the  value  of  (65)  by  u ;  and  hence  (66)  becomes 

V  =  u\  (67) 

If  we  now  return  to  (63),  and  express  the  value  of  z  in  polar 
coordinates,  we  have 


(since  #2  +  y2  —  r2).  The  elementary  area  in  the  plane  2  =  0 
becomes  rdO .  dr  instead  of  dxdy,  and  the  limits  of  the  inte- 
gration (in  order  to  include  the  same  part  of  the  solid  as 


APPENDIX.  199 

before)  are  0  and  oo  for  r,  and  0  and  ^-TT  for  0.     Hence  the 
volume  of  the  solid,  expressed  in  polar  coordinates,  is 

y=  T  r\-r* rdOdr.  (68) 

0         0 

Integrating  with  respect  to  6  we  have 

00 

V  =  ^  r<rr*rdr.  (69) 


This  integral  has  the  same  form  as  the  first  of  the  expressions 
in  (62),  and  hence  we  have 


l)  =  i;  (70) 

and  therefore,  from  (69),  ;. 


Equating  this  value  of  V  with  that  given  in  equation  (67), 
we  have 

tta=j>         or  M  =  jVi  =  0.8862269..., 

which  is  therefore  the  value  of  (65). 
Returning  now  to  the  integral 


/ 


(71) 


let  us  make  successive  applications  of  equation  (61),  until 
the  exponent  of  x  has  been  reduced  to  either  unity  or  zero 
(according  as  n  is  odd  or  even).  We  shall  then  have  (71) 
expressed  as  a  series,  whose  terms,  with  the  exception  of  the 
last  one,  will  all  be  of  the  form 

Axm  e~< 


200  APPENDIX. 

Evidently  this  expression  is  zero  when  x  =  0,  and  it  can  be 
shown,  by  the  usual  methods  employed  in  the  differential 
calculus  for  evaluating  indeterminate  quantities,  that  it  is 
also  zero  when  x  =  oo  .  Hence  when  (71)  is  integrated 
between  the  limits  0  and  oo  all  the  terms  of  the  series 
obtained  by  successive  applications  of  (61)  disappear,  with 
the  exception  of  the  last  one,  which  is 

~  f<r**xdx  (72) 

0 

when  n  is  odd,  and 

-J-  -f<-*'dx  (73) 

0 

when  n  is  even.  We  have  already  found  that  the  value  of 
the  integral  in  (72)  is  £  [see  equation  (70)],  and  that  the 

value  of  that  in  (73)  is  — •     Hence,  finally, 


f)     2 

0 


...   ,.f      .       ... 

—  (^  n  is  odd), 


...    r  r*      •  N 

—  VTT  (if  n  is  even). 


2  2 


(74) 


These  expressions  have  several  applications  in  the  text.  If, 
for  example,  we  put  v  =  ax  in  the  equation  immediately 
following  (3),  on  page  25,  we  have 


2Nma*          x* 

= -=-   I  €~*  x*dx. 

VTT     4 


Here  the  exponent  of  x  is  even,  and  by  applying  the  second 
of  the  values  given  in  (74)  we  have 

.       2Nma?    3X1      ,-      3 
7r  =  - 


APPENDIX.  201 

as  in  equation  (4).      Again,   if  we  put  v  =  ax,  the  second 
expression  after  (1),  on  page  24,  becomes 


/. 


<"  xsdx. 


Here  the  exponent  of  x  is  odd,  and  applying  the  first  of  the 
values  given  in  (74)  we  find  that  the  expression  under  con- 
sideration becomes 

2 


and  hence  F"0  =  —  -=, 

VTT 

as  in  equation  (2). 

In  order  to  integrate  (1)  between  given  finite  limits,  we 
may  first  put  v  =  ax,  as  before,  and  then  expand  the  expo- 
nential function  in  accordance  with  the  formula 

w2  wz 

<«=!+„,+_+__+..., 

after  which  the  integration  may  be  performed  term  by  term. 
The  series  thus  obtained  is 


1560  +  10800      '"/' 

Performing  the  integration  between  the  limits  v  =  Q  and 
v  =  mVo  (see  page  26)  is  the  same  thing  as  performing  it 

between  the  limits  x  =  0  and  x  =  ^—^  (  since  x  =  -  Y     By 

a     ^  a) 

referring  back  to  equation  (2)  we  see  that 
mV0_2m 
*      ~V^5 

and  it  was  by  substituting  this  quantity  in  (75)  that  the  table 
on  page  26  was  calculated  for  the  values  m  =  £  to  m  =  1, 
inclusive.  When  m  is  much  greater  than  unity  (75)  con- 


202  APPENDIX. 

verges  so  slowly  that  it  is  no  longer  useful  for  calculation. 
For  higher  values  of  m  we  therefore  develop  the  integral 
in  (1)  in  a  different  manner.  If  we  first  put  v  =  ax,  as 
before,  and  then  integrate  by  parts  once,  in  accordance  with 
(61),  the  integral  of  equation  (1),  between  the  limits  0  and  x, 
becomes 


(76) 

Now  we  have,  in  general, 

oo  oo 

Cxdx  =  Cxdx  —  Cxdx, 

0  0  x 

where  X  is  any  continuous  function  of  x  ;  and  hence 


/ 

r 


*2  dx  =         *1  dx  -      t'**  dx.  (77) 

r 


By  substituting  in  (76)  and  remembering  that  the  value  of 
the  first  integral  in  the  second  member  of  (77)  is  -jV^,  the 
integral  of  (1),  between  the  limits  0  and  x,  becomes 


-  *<-*'  -          °  dx     •  (78) 


By  successive  applications  of  (61),  we  may  develop  the  integral 
in  (78)  in  terms  of  descending  powers  of  x.     Thus  : 


(79) 


This  series  is  apparently  divergent,  but  it  can  easily  be  shown 
that  the  sum  of  all  the  terms  after  the  nth  term  is  less  than 
the  nth  term  multiplied  by  e~x  ;  and  hence,  in  applying  (79), 
we  should  compute  the  series  until  its  terms  begin  to  increase; 
and  if  we  preserve  the  terms  only  up  to  this  point  the  result 
will  be  very  close  to  the  true  value  of  the  integral,  when  x  is 


APPENDIX. 

^s^UAUK  V£^r 

large.     By  substituting  (79)  for  the  integral  in  (78),  the  inte- 
gral of  (1)  between  the  limits  0  and  x  is  found  to  be 

1         1         3         15 


This  expression  converges  rapidly  for  large  values  of  x,  and  it 
was  used  in  computing  the  values  given  in  the  table  on  page 
26  for  m  =  2,  3,  and  4,  and  also  for  computing  the  numbers 
mentioned  on  page  27.  As  explained  above,  the  value  of  x 

O  ™ 

corresponding  to  any  given  value  of  m  is  —  •=• 

V-7T 

Rankine's  Method  for  Calculating  the  Ratio  of  the  Specific 
Heats  of  Gases.  —  It  is  not  easy  to  measure  the  specific  heat 
of  a  gas  at  constant  volume,  and  hence  the  ratio  of  the  specific 
heats  is  usually  obtained  by  some  indirect  process.  If  the 
specific  heat  at  constant  pressure  is  known,  this  ratio  may  be 
determined  by  the  following  method.  Conceive  a  unit  weight 
of  gas  to  be  confined  in  a  vertical  cylinder  provided  with  a 
piston.  Let  Vl  be  the  volume  of  the  gas,  and  ^  its  absolute 
temperature.  With  the  piston  fixed  (so  that  Vi  cannot  change) 
let  the  gas  be  heated  until  its  temperature  becomes  t2.  The 
quantity  of  heat  required  to  effect  this  change  will  be 


where  Sv  is  the  specific  heat  of  the  gas  at  constant  volume; 
and  multiplying  this  expression  by  J  (the  mechanical  equiva- 
lent of  heat)  we  find  that  the  mechanical  energy  required  to 
effect  the  change  in  temperature  is 

e/S,(*.-*i).  •  (80) 

Now  let  us  conceive  that  instead  of  heating  the  gas  at  con- 
stant volume,  we  raise  its  temperature  through  the  same  range 
as  before,  but  with  the  pressure  of  the  gas  constant.  This 
can  be  effected  by  loading  the  piston  with  a  known  weight, 
and  allowing  the  gas  to  expand  by  pushing  the  weight  before 


204  APPENDIX. 

it.  The  amount  of  energy  required  to  raise  the  temperature 
of  the  gas  from  ^  to  tz  under  the  new  conditions  will  be 

JSp(t,-tJ,  (81) 

where  Sp  is  the  specific  heat  of  the  gas  at  constant  pressure. 
Now  (81)  will  be  greater  than  (80),  because  the  gas,  in 
expanding,  does  work  in  raising  the  weight.  Moreover,  a 
certain  amount  of  energy  is  required  to  separate  the  molecules 
of  the  gas  against  the  attractions  that  they  exert  upon  one 
another.  The  experiments  of  Joule  and  Thomson  have  shown 
that  this  last  quantity  is  very  small  for  the  "permanent 
gases,"  and  in  the  present  calculation  we  shall  neglect  it, 
and  assume  that  the  difference  between  (80)  and  (81)  is  due 
solely  to  the  external  work  that  the  gas  does  in  pushing  up 
the  weight.  If  the  weight  is  such  that  the  gas  exerts  a 
pressure  P  against  each  unit  area  of  the  cylinder  and  piston, 
the  external  work  done  will  be 


where   Fj  and   F2  are  the  initial  and  final  volumes,  respec- 
tively.    Hence,  returning  to  (80)  and  (81),  we  have 

JSP  ft  -  *i)  =  JSV  ft  -  *i)  +  P  (  F2  -  FO,  (82) 

or,  dividing  by 


Now  the  characteristic  gas-equation  gives  us 
PV^  =  Rtz     and    PY1  =  Et1 
and  hence,  subtracting  and  dividing  by  (£2  — 


ft-*) 

Making  this  substitution  in  (83),  we  have 


(84) 


APPENDIX. 


205 


from  which  the  ratio  of  the  specific  heats  can  be  determined, 
if  either  of  them  is  known.  Thus  if  Sp  is  known,  we  have, 
from  (84), 

S,,      .        R 


s 


(85) 


To  compute  R  for  any  given  gas  we  must  know  the  volume 
(  F0)  of  a  unit  weight  of  the  gas  at  some  definite  temperature 
and  pressure  —  say  at  TO  and  PQ.  Then 


and  (85)  becomes 


S 


(86) 


(87) 


The  computation  is  given  below,  in  tabular  form,  for  five  of 
the  so-called  "permanent  gases."  In  this  table  F0  is  the 
volume  of  a  gramme  of  the  gas  (expressed  in  cubic  centi- 
meters), this  volume  being  measured  at  the  freezing  point  of 
water  (TO  =  273.1°  C.),  and  under  a  pressure  (P0)  of  1033.3 
grammes  per  square  centimeter.  I  have  used  Grifnths's  value 
of  J";  that  is,  I  have  assumed  that  4.27  X  10 4  centimeter- 
grammes  of  energy  will  raise  the  temperature  of  one  gramme 
of  water  from  0°  C.  to  1°  C. 

Of 

COMPUTATION  OF  -^  FOR  IT,  0,  JV,  CO,  AND  C02. 


PoF0 

GAS 

Fo 

T()J 

Sp 



Ov 

8, 

Hydrogen  .     .     . 

11,157.6 

0.9887 

3.4090 

0.2900 

0.7100 

1.408. 

Oxygen      .     .     . 

699.0 

0.0619 

0.2175 

0.2848 

0.7152 

1.39& 

Nitrogen     .     .     . 

795.6 

0.0705 

0.2438 

0.2893 

0.7107 

1.40T 

Carbon  monoxide 

809.6 

0.0717 

0.2438 

0.2942 

0.7058 

1.417' 

Carbon  dioxide    . 

505.4 

0.0448 

0.2169 

0.2065 

0.7935 

1.260* 

I  have  also  computed  the  ratio  of  the  specific  heats  of 
certain  vapors  by  the  same  process  ;  and  although  we  cannot 


206  APPENDIX. 

properly  neglect  the  effects  of  inter-molecular  attractions  in 
such  cases,  the  results  obtained  by  applying  equation  (82)  will 
be  sufficiently  accurate  for  the  purpose  for  which  we  require 
them  —  namely,  for  computing  the  value  of  K9  in  equation 
(41).  The  ratios  of  the  specific  heats,  as  obtained  by  this 
process  for  the  first  three  of  the  vapors  in  the  table  on 
page  101,  are  as  follows  :  For  water-vapor,  y  =  1.296  j  for 
alcohol,  y  =  1.103;  and  for  bisulphide  of  carbon,  y  =  1.198. 
For  mercury  vapor  I  have  used  the  value  of  y  given  by  the 
experiments  of  Kundt  and  Warburg  on  the  velocity  of  sound 
in  that  vapor  —  namely,  y  =  1.66. 

Plateau's  "Liquide  Glyce*rique."  —  Following  is  a  trans- 
lation of  M.  Plateau's  observations  on  this  subject :  *  "  The 
films  obtained  from  a  simple  solution  of  soap  have  very  little 
persistence  unless  they  are  protected  by  a  glass  shade.  A 
soap  bubble  four  inches  in  diameter  rarely  lasts  two  minutes 
in  the  free  air  of  a  room,  and  usually  it  bursts  in  one  minute, 
or  even  in  half  a  minute.  It  was  therefore  important  ...  to 
discover  some  better  liquid ;  and  in  following  out  an  idea 
suggested  by  M.  Donny,  I  have  been  fortunate  enough,  after 
a  number  of  trials,  to  obtain  a  liquid  which  gives  films  of 
remarkable  persistence.  It  consists  in  a  mixture  of  glycerine 
and  a  solution  of  soap,  and  I  call  it  the  ' glycerine  liquid'.  .  . 
Let  us  describe  the  preparation  of  the  liquid.  In  the  first 
place,  it  ought  to  be  prepared  in  summer,  at  a  time  when  the 
temperature  of  the  room,  during  the  day,  at  least,  does  not 
fall  below  20°  C.  [68°  Fahr.]  ;  for  at  temperatures  much  lower 
than  this  the  results  are  poor.  Marseilles  soap  is  recommended, 
and  my  experience  indicates  that  the  best  glycerine  for  the 
purpose  is  that  made  in  England  and  known  a"s  <  Price's 
glycerine7.  I  shall  assume,  in  what  follows,  that  these  sub- 
stances are  used,  and  that  the  operations  are  conducted  at  the 
proper  temperature.  It  is  not  impossible  to  succeed  with 

*  Plateau,  Statique  Exptrimentale  et  Theorique  des  Liquides,  Vol.  I, 
page  161. 


APPENDIX.  207 

other  soaps  and  other  glycerines,  but  then  the  proportions 
must  be  changed,  and  I  can  give  no  general  rule  for  them. 
The  process  of  preparation  will  vary  somewhat,  according  to 
what  it  is  desired  to  accomplish.  In  the  first  place,  if  the 
experimenter  is  more  desirous  of  simplicity  of  preparation 
than  of  excellence  in  the  results,  he  may  proceed  as  follows  : 
Fresh  Marseilles  soap,  which  has  retained  all  its  moisture,  is 
scraped  into  shavings  and  is  dissolved  with  gentle  heat  in 
distilled  water,  forty  parts  (by  weight)  of  water  being  allowed 
to  each  part  of  soap.  When  the  solution  has  cooled  to  about 
the  temperature  of  the  room,  it  is  filtered  until  the  filtrate  is 
clear.  Three  volumes  of  the  clear  soap  solution  are  then  put 
in  a  flask,  two  volumes  of  Price's  glycerine  are  added,  and  the 
flask  is  shaken  violently  until  its  contents  are  thoroughly 
mixed.  After  this,  it  is  set  aside  until  the  next  day.  Then, 
according  to  the  quality  of  the  Marseilles  soap,  it  may  be 
found  that  the  liquid  is  still  limpid,  or  that  it  contains  a 
heavy  precipitate.  In  the  first  case  the  liquid  is"  ready  for 
use,  and  the  maximum  duration  of  a  four-inch  bubble  blown 
with  it  will  be  about  an  hour  and  a  half  ;  but  the  liquid  will 
gradually  lose  its  property  of  yielding  persistent  films,  and 
after  a  fortnight  a  four-inch  bubble  blown  with  it  will  last 
only  about  ten  minutes.  In  the  second  case  the  precipitate, 
at  first  held  in  suspension  throughout  the  liquid,  will  rise 
toward  the  surface  with  extreme  slowness,  and  after  some 
days  it  will  gather  into  a  layer  sharply  separated  from  the 
rest  of  the  solution.  The  clear  liquid  is  then  drawn  off  by 
means  of  a  siphon  provided  with  a  side  tube  for  starting  the 
flow,  and  the  preparation  is  completed.  I  ought  to  add  that 
when  the  short  branch  of  the  siphon  is  introduced,  a  portion 
of  the  deposit  is  carried  down  by  it,  and  adheres  to  the 
exterior  surface  of  the  tube  in  the  form  of  a  sort  of  reversed 
cone ;  and  before  filling  the  siphon  this  adherent  deposit  should 
be  removed.  To  effect  the  removal  the  apparatus  should  first 
be  left  to  itself  for  a  quarter  of  an  hour,  after  which  the 
siphon  should  be  slightly  agitated  in  a  horizontal  direction. 


208  APPENDIX. 

The  cone  of  deposit  will  then  come  away  in  little  lumps, 
which  gradually  float  upward  and  join  the  layer  of  deposit 
above.  The  liquid  obtained  in  this  way  is  much  better  than 
the  preceding  one.  It  is  ready  for  use  as  soon  as  drawn 
off  by  the  siphon,  and  four-inch  bubbles  blown  with  it  have  a 
maximum  persistence  of  three  hours.  It  will  remain  in  good 
condition  for  nearly  a  year.  ...  If  the  experimenter  is  willing 
to  devote  more  care  to  the  process  of  preparation,  a  far  better 
liquid  may  be  obtained  in  the  following  manner  :  After  having 
prepared  the  solution  of  soap  as  before,  15  volumes  of  it  are 
intimately  mixed  with  11  volumes  of  Price's  glycerine,  and 
the  resulting  liquid  is  allowed  to  stand  for  seven  days. 
During  this  time  a  precipitate  may  form,  or  the  liquid  may 
remain  clear,  according  to  the  quality  of  the  soap ;  but  the 
subsequent  treatment  is  the  same  in  either  case.  On  the 
morning  of  the  eighth  day  the  flask  containing  the  mixture  is 
immersed  in  water  which  has  been  cooled  to  about  3°  C.  by 
shaking  it  with  small  pieces  of  ice,  and  the  temperature  is 
maintained  at  this  point  for  six  hours  by  adding  ice  from 
time  to  time.  If  the  mixture  of  glycerine  and  soap  solution 
is  to  be  prepared  in  any  considerable  quantity,  it  is  better  to 
divide  it  among  several  flasks,  in  order  that  it  may  sooner 
acquire  the  temperature  of  the  bath.  During  the  prolonged 
action  of  the  cold,  an  abundant  precipitate  will  be  produced. 
When  the  six  hours  have  elapsed,  the  liquid  is  filtered 
through  a  rather  porous  filter-paper,  and  if  there  is  much  of 
it,  several  filters  should  be  used  simultaneously.  It  is 
necessary  to  prevent  the  liquid  in  the  filters  from  becoming 
warmed,  for  otherwise  the  precipitate  produced  by  the  cold- 
bath  would  partially  dissolve.  For  this  purpose,  and  before 
beginning  the  filtration,  a  small,  elongated  bottle,  filled  with 
fragments  of  ice,  is  carefully  placed  in  each  filter,  and  closed 
with  a  glass  stopper  in  order  that  it  may  have  sufficient 
weight.  This  bottle  should  be  so  inclined  that  its  side  may 
rest  against  the  filter,  and  the  bottoms  of  all  the  flasks  that 
contain  the  filter-funnels  must  also  be  surrounded  by  frag- 


APPENDIX.  209 

ments  of  ice.  The  soap  solution  is  then  removed  from  the 
cold-bath  and  poured  immediately  into  the  niters.  The  first 
portions  of  the  filtrate  will  contain  a  precipitate,  and  they 
should  therefore  be  poured  back  into  the  filters  again.  After 
this  has  been  repeated  two  or  three  times  the  filtrate  will 
become  perfectly  clear.  It  is  hardly  necessary  to  add  that  if 
the  filtration  requires  a  considerable  time,  it  may  be  necessary 
to  renew  the  ice  in  the  little  bottles  from  time  to  time.  So 
far  as  the  ice  about  the  bottoms  of  the  flasks  is  concerned,  it 
may  be  said  that  the  only  purpose  that  it  subserves  is  to 
prevent  the  first  portions  of  the  filtrate  (which  are  poured 
back  upon  the  filters)  from  becoming  warm,  and  that  it  is  no 
longer  necessary  when  the  filtrate  has  become  clear.  When 
the  filtration  is  complete,  the  filtrate  is  allowed  to  stand  for 
ten  days,  after  which  it  is  ready  for  use.  With  a  liquid 
prepared  in  this  manner,  a  four-inch  bubble,  under  the  most 
favorable  conditions,  may  last  as  long  as  eighteen  hours." 
Marseilles  soap  is  a  mixture  of  oleate,  stearate,  and  margarate 
of  soda,  and  Plateau  considered  that  the  precipitate  thrown 
down  by  cold  consists  of  the  last  two  of  these  substances.  He 
found  that  if  pure  oleate  of  soda  is  used  in  place  of  the  soap, 
the  precipitate  is  not  formed ;  and  he  obtained  a  four-inch 
bubble,  from  an  oleate  solution,  which  lasted  over  twenty-four 
hours.  Plateau  insists  strongly  on  the  proportions  that  he 
gives. 

On  the  Thermal  Phenomena  Produced  by  Extending  a 
Liquid  Film.  —  In  the  Proceedings  of  the  Koyal  Society  for 
1858,  Lord  Kelvin  has  shown  that  certain  thermal  changes 
occur  when  the  area  of  a  liquid  film  is  increased ;  and  on 
pages  144  and  145  of  the  present  volume  these  changes  are 
taken  into  account,  in  computing  the  sizes  of  molecules  by  the 
surface  tension  method.  It  will  be  convenient,  in  investigating 
these  thermal  phenomena,  to  consider  the  behavior  of  a  liquid 
film  —  such  a  film,  for  example,  as  that  shown  in  Fig.  33. 
Let  us  suppose  that  at  a  given  instant  the  absolute  temperature 


210  APPENDIX. 

of  this  film  is  T,  the  surface  tension  being  S,  and  the  total 
area  of  the  film  (counting  both  sides)  being  A.  It  is  known 
that  S  varies  with  T,  and  it  is  also  known  that  so  long  as  r 
remains  constant,  S  does  not  vary  with  A.  It  may  be,  however, 
that  r  tends  to  vary  with  A  ;  and  in  such  a  case,  unless  special 
care  were  taken  to  maintain  a  constant  temperature  by  the 
addition  or  abstraction  of  heat  from  without,  we  should  find 
an  apparent  connection  between  S  and  A,  when  no  direct 
connection  between  them  really  exists.  In  studying  the 
thermal  phenomena  of  films  we  have  therefore  to  consider 
two  general  cases  :  (1)  when  A  varies  adiabatically  —  that  is, 
without  absorbing  or  giving  up  heat,  as  such,  to  external 
bodies,  —  and  (2)  when  A  varies  isothermally,  its  constancy  of 
temperature  being  maintained  by  the  addition  or  abstraction 
of  heat  by  means  of  some  external  agency.  Let  us  first 
consider  the  adiabatic  change  of  A.  Assuming  that  the 
temperature,  T,  does  vary  during  adiabatic  extension,  it  is 
plain  that  the  increase  of  T,  per  unit  increase  of  A,  is  rep- 


( •rj  J  j 


resented  by  the  differential  coefficient     •rj  J  j  wnose  value,  as 

yet,  is  quite  unknown.  This  being  the  increase  of  r  per  unit 
increase  of  A,  it  is  also  plain  that  when  A  increases  by  an 
amount  dA}  the  increment  of  T  will  be 


and  the  amount  of  energy  appearing  as  sensible  heat  will  be 

A,  (89) 

where  h  is  the  quantity  of  energy  required  to  raise  the 
temperature  of  the  entire  film  1°.  If,  now,  we  wished  to 
extend  the  film  isothermally,  we  should  have  to  subtract  from 
it,  for  each  dA  of  increase  in  area,  the  quantity  of  heat-energy 
that  is  represented  by  (89).  The  amount  of  mechanical 
energy  expended  upon  the  film  in  extending  it  an  amount 
dA  against  the  surface  tension,  S,  would  be 

S-dA;  (90) 


APPENDIX.  211 

and  hence  the  total  increase,  dw,  of  the  energy  of  the  film 
would  be  the  difference  between  (90)  and  (89),  and  we  should 
have 


dw  =  S-dA  —  h-    j2    -  dA. 
Upon  dividing  this  expression  by  dA  we  have 


£="-*•(£) 


which  is  the  total  increase  in  the  energy  of  the  film,  per  unit 
increase  of  A.  The  differential  coefficient  in  the  right-hand 
member  of  (91)  might  be  found  by  direct  experiment,  but  it 
is  more  convenient  to  express  it,  by  Lord  Kelvin's  method,  in 
terms  of  another  quantity  whose  value  can  be  found  with 
less  difficulty.  To  effect  the  transformation  in  question, 
consider  the  film  in  Fig.  33  to  be  the  working-body  of  a  heat 
engine,  and  conceive  it  to  pass  through  the  following  infini- 
tesimal cycle :  (1)  It  is  extended  by  an  amount  A^4,  the  heat 
produced  during  this  extension  being  removed  so  that  the 
process  is  isothermal ;  (2)  the  film  is  then  further  extended, 
adiabatically,  by  an  infinitesimal  amount,  dA,  the  temperature 
increasing,  at  the  same  time,  by  an  amount  dr ;  (3)  the  film 
is  then  allowed  to  contract,  isothermally,  to  such  a  point  that 
the  fourth  stage  in  the  cycle  shall  cause  it  to  return  to  its 
initial  condition ;  (4)  it  is  then  allowed  to  contract  adiabati- 
cally, until  the  cycle  is  completed.  If  a  diagram  of  this 
cycle  be  drawn  on  paper,  with  film-areas  as  abscissae  and 
surface-tensions  as  ordinates,  the  isothermals  will  be  straight 
lines  parallel  to  the  axis  of  abscissae,  and  the  cycle,  being 
infinitesimal,  may  be  regarded  as  a  parallelogram.  The  length 
of  the  parallelogram  will  be  A^4,  and  its  height  will  be 

~y<*r, 


this  being  the  change  which  £  undergoes  when  the  temperature 
increases  by  dr.    The  amount  of  mechanical  energy  gained  in 


212  APPENDIX. 

the  cycle  will  be  represented  by  the  area  of  the  parallelogram; 
that  is,  it  will  be 

^A'(^\dr.  (92) 

\drj 

The  quantity  of  heat  abstracted  from  the  film  during  the 
first  stage  of  the  cycle  is 

Ai_\ 

(93) 

as  we  see  by  comparison  with  equation  (88) ;  and  the  quantity 
added  to  it  during  the  third  stage  differs  from  (93)  by  only 
an  infinitesimal  amount.  The  thermodynamic  efficiency,  E, 
of  the  film,  is  therefore  found  by  dividing  (92)  by  (93) ;  and 
we  have 

par 

js=±-4-.^- 

dA 

But  since  the  cycle  under  consideration  is  reversible,  the 
efficiency  must  also  be  expressible  by  "Carnot's  formula"; 
and  hence 


Upon  equating  these  two  values  of  E  we  find  that 

(dr\_r   (dS\ 
\dAJ~h'  \dr)' 

Substituting  this  in  (91)  we  find  that  the  total  increase  in  the 
energy  of  the  film,  per  unit  of  isothermal  increase  in  area,  is 


This  may  be  written  in  the  form 

T    d& 
~ls'd~7 


APPENDIX. 


213 


where  the  parenthesis  is  the  quantity  represented  by  m  on 
page  144.  The  computation  of  m  for  the  four  liquids  dis- 
cussed on  page  145  is  presented  below  in  tabular  form. 


COMPUTATION  or  m  FOR  VARIOUS  LIQUIDS,  AT  20°  C. 


LIQUID 

8 

dS 

dr 

m 

Water 

074 

000143 

1  57 

Alcohol    

.026 

—.000121 

2  36 

Bisulphide  of  Carbon 

033 

.000131 

2  16 

Mercury  

.551 

—.00032 

1.17 

The  value  of  —  for  water  is  given  by  the  experiments  of 

Mr.  T.  Proctor  Hall,  mentioned  on  page  95  of  this  volume. 
The  corresponding  values  for  alcohol,  bisulphide  of  carbon, 
and  mercury,  were  obtained  by  assuming  (1)  that  the  surface 
tension  is  a  linear  function  of  the  temperature,  and  (2)  that  it 
vanishes  at  the  critical  temperature,  where  the  liquid  becomes 
indistinguishable  from  its  vapor.  The  critical  temperature 
of  mercury  is  not  known;  but  by  comparing  the  specific  heat 
of  this  liquid  with  the  value  of  W  given  on  page  101,  the 
critical  temperature  was  inferred  to  be  in  the  neighborhood 
of  1,700°  C. 


INDEX. 


Absolute  thermometer  scale,  45, 46. 

zero,  46. 

Acetic  acid,  critical  temperature  of, 

84. 

Adams,  163. 
Air,  constant  composition  of,  31. 

thermometer,  45,  46. 

ratio  of  specific  heats  of,  52. 

cooling   effect   of   expansion 

through  porous  plug,  54. 

coefficient  of  viscosity  of,  63. 

free  path  of  molecules  of,  66. 

Alcohol,  critical  temperature  of,  84. 

influence  of,  on  surface  ten- 
sion of  water,  89. 

surface  tension  of,  94. 

latent  heat   of  vaporization 

of,  96. 

work  required  to  form  more 

surface,  101. 

size  of  molecules  of,  145. 

vapor,  ratio  of  specific  heats 

of,  206. 

Amagat,  136. 

Ammonia,  critical  temperature  of, 

84. 

Ampere,  5. 
Andrews,  58. 
Assumptions   of    original    kinetic 

theory,  22. 

of  generalized  theory,  34. 

provisional,  concerning  con- 
stitution of  molecules,  171. 

Atoms  and   molecules,   difference 
between,  6. 


Attraction,  molecular,  in  gases,  53, 
57. 

law  of,  54,  55. 

possible  relation  to  gra- 
vitation, 56. 

in  liquids,  74. 

work  done  against,  in 

forming  liquid  surface,  101. 

range  of,  by  Quincke's 

method,  145. 

necessity  of  explaining, 

153. 

Avogadro's  hypothesis,  5, 6,  41,  43, 
148. 

Ball,  elastic,  behavior  of,  18. 
Balmer's  law,  186. 
Barium,  atomic  weight  of,  162. 
Bees,  analogy  of  molecules  with,  15. 
Benzene,  critical  temperature  of, 

84. 

Bode's  law,  163. 
Body,  rigid,  degrees  of  freedom  of, 

33. 

Boiling,  phenomena  of,  80. 
Boltzmann's  law,  35,  37,  155,  188. 
Boscovich,  178. 
Boyle's  law,  41,  43. 
Bromine,  atomic  weight  of,  162. 
Brunner,  95. 
Bullets,  analogy  of  molecules  with, 

17. 
Burton,  Dr.  C.  V.,  180,  195. 

Calcium,  atomic  weight  of,  162. 


216 


INDEX. 


Camphor,  90,  142. 
Capillary  curve  of  water,  91. 

phenomena ;  —  see  Surface 

tension. 

Carbon,  Dalton's  symbol  for,  3. 
Carbon  dioxide,  composition  of,  2. 
Dalton's  symbol  for,  3. 

velocity  of  molecules,40. 

ratio  of  specific  heats  of, 

52,  205. 

degrees  of  freedom  of 

molecules  of,  52. 

cooling  effect  of  expan- 
sion through  porous  plug,  64. 

Clausius' corrected  equa- 
tion for,  57. 

coefficient   of   viscosity 

of,  63,  65. 

free  path  of  molecules 

of,  66. 

molecular  collisions  in, 

67. 

critical  temperature  of, 

84. 

aggregate  volume  of  mol- 
ecules of,  135. 

maximum    density    of, 

136. 

size  of  molecules  of,  139. 

number  of  molecules  per 

unit  volume,  148. 

and  Dulong  and  Petit' s 

law,  157. 

Carbon   disulphide,   critical    tem- 
perature of,  84. 

surface  tension  of,  94. 

latent  heat  of  vaporiza- 
tion of,  96. 

work  done  in  forming 

more  surface,  101. 

-  size  of  molecules  of,  145. 

ratio  of  specific  heats  of 

vapor,  206. 


Carbon  monoxide, composition  of,  2. 

Dalton's  symbol  for,  3. 

velocity  of  molecules,40. 

degrees   of    freedom  of 

molecules  of,  48. 

-  ratio  of  specific  heats  of, 

49,  205. 
coefficient   of    viscosity 

of,  63. 
free  path  of  molecules 

of,  66. 
molecular  collisions  in, 

67. 
and  Dulong  and  Petit's 

law,  157. 

Charles,  law  of,  43. 
Chlorine  a  diatomic  gas,  10. 

critical  temperature  of,  84. 

and  Prout's  hypothesis,  159. 

atomic  weight  of,  162. 

Chloroform,  critical  temperature  of, 

84. 

surface  tension  of,  94. 

Chree,  183. 

Classification  of  bodies,  10. 

Clausius,  52,  56,  64,  66,  120,  134, 

138. 

Cleavage  in  crystals,  132. 
Collisions,  molecular,  in  liquids,  14. 
in  gases,  15,  20,  67. 

of  cubes,  22. 

of  spheres,  23. 

Comet,  Encke's,  retardation  of,  166. 
Compounds  vs.  elements,  161. 
Compressibility  of  liquids,  84. 
Constitution  of  solids,  12,  103, 106, 
127. 

of  liquids,  12,  74. 

of  gases,  14. 

of  molecules,  53,  151,  171. 

Copper  and  zinc,  size  of  molecules 

of,  140. 
Critical  temperatures,  82. 


INDEX. 


217 


Critical  temperatures,  table  of,  84. 

velocities  of  molecules,  50,  82. 

Crookes,  62,  74. 

Crookes's  tubes,  70. 

Crystals,  physical  properties  of,  124. 

bounding  planes  of,  125. 

rationality  of  indices  of,  126, 

131. 

molecular  structure  of,  127. 

cleavage  in,  132. 

Cubes,  collision  of,  22. 

Dalton,  2. 

Dalton's  laws,  2,  40,  43. 

Davy,  173. 

Degrees  of  freedom,  32,  36. 

partition  of  kinetic  en- 
ergy among,  35,  37,  157,  188. 

Densities  and  molecular  weights  of 
gases,  relation  of,  5. 

of  gases,  maximum,  136. 

Density,  vapor,  77. 
Deville,  110,  112,  120. 
Dewar,  137. 

Diffusion  in  gases,  58,  66. 

in  solutions,  114. 

Dissociation,  9,  50,  110. 
Distillation,  122. 
Donny,  206. 

"  Doughnut  theory,"  180. 

Dufour,  81. 

Dulong,  52. 

Dulong  and  Petit' s  law,  154. 

Ebullition,  phenomena  of,  80. 
Ekaboron,  163. 

Elasticity,  possibility  of  explaining, 
153,  172,  177. 

of  molecules,  15,  17,  20,  172, 

177. 

Electrolysis,  120. 

Electro-magnetic  theory  of  light, 
168. 


Elements,  10,  161. 
Encke's  comet,  166. 
Energy,     transformations    of,     in 
elastic  ball,  18. 

kinetic,  partition  of,  in  mol- 
ecules, 31,  35,  37,  122,  155,  157, 
188. 

vibratory,  in  molecules,   19, 

36,  182. 

minimum  potential,  105,  113, 

114,  127. 

"English,"  23. 

Equations,  adaptation  of,  to  gen- 
eralized kinetic  theory,  37. 

of  Vander  WaalsandClausius, 

56. 

the    integration    of    certain, 

197. 

Equilibrium,    chemical,    Guldberg 

and  Waage's  theory  of,  112. 
Ether,  critical  temperature  of,  84. 

effect  of,  on  the  surface  ten- 
sion of  water,  89. 

Ether,  the  luminiferous,  160,  164. 

Rankine's  theory  of ,  174. 

Dr.  Burton's  theory  of, 

180. 
"  Ether  squirt "  and  "  Ether  sink  " 

theories  of  gravitation,  191,  192. 
Evaporation,  free,  75. 

cooling  effect  of,  77. 

latent  hesat  of,  95. 

"Explanation,"   meaning    of   the 

term,  153. 


Faraday,  168. 

Films,  liquid,  phenomena  of,  87. 

thermal  phenomena  of, 

144,  209. 

Fluid,  definition  of,  11. 
Forces,  interrnolecular,  49,  53,  74. 
Free   path   in    gases,    14,    15,   66, 

68. 


218 


INDEX. 


Freedom,  degrees  of,  32,  36. 
partition  of  kinetic  en- 
ergy among,  35,  37,  157,  188. 
Friction,  internal,  in  gases,  60. 

Gas,  definition  of,  11. 

equations  of  Van  der  Waals 

and  Clausius,  56. 

number  of  molecules  in  a  unit 

volume  of,  148. 

Gases,  volumetric  relations  of,  5. 

dissociation  in,  9,  50. 

molecular  constitution  of,  14. 

kinetic  theory  of,  14. 

free  paths  of  molecules  of,  14, 

15,  66. 

diagram  of  molecular  motion 

in,  16. 

pressure  in,  17,  38. 

objections  to  kinetic  theory 

of,  19,  49. 

molecular  velocities  in,  20, 29, 

39,  40. 

fundamental  assumptions  of 

original  kinetic  theory  of,  22. 

distribution  of  velocities  in,  24. 

properties  of  mixtures  of,  31, 

34,  40. 

assumptions    of    generalized 

kinetic  theory  of,  34. 

partition  of  kinetic  energy  in, 

31,  35,  37,  188. 

kinetic  theory  of,  compared 

with  observation,  42. 

objections  to,  49,  52. 

ratio  of  specific  heats  of,  47, 

203. 

expansion  of,  through  porous 

plug,  53. 

molecular  attraction  in,  53, 

57. 

diffusion  in,  58,  66. 

viscosity  of,  59. 


Gases,  viscosity  of,  kinetic  explana- 
tion of,  63. 

molecular  collisions  in,  67. 

critical  points  of,  84. 

maximum  density  of,  136. 

Gay  Lussac,  law  of,  43. 
Glycerine  liquid,  Plateau's,  206. 
Graham,  4,  62. 

Gravitation,  vibratory  theory  of, 
166,  192. 

Newton's  theory  of,  190. 

Le  Sage's  theory  of,  190. 

"  ether  squirt"  theory  of,  191. 

-  "  ether  sink  "  theory  of,  192. 

-  Maxwell's  theory  of,  193. 

vortex  theory  of,  194. 

Preston's  vortex  filament  the- 
ory of,  194. 

Dr.  Burton's  strain-figure  the- 
ory of,  195. 

Griffiths,  52,  205. 
Guldberg  and  Waage,  112. 

Hagen,  91. 

Hall,  T.  Proctor,  95,  213. 

Heats,  specific,  of  gases,  ratio  of, 

47,  203. 

Helmholtz,  176. 
Hertz,  170. 
Holman,  65. 
Hydrochloric  acid,  surface  tension 

of,  94. 
and  Dulong  and  Petit's 

law,  157. 
Hydrogen,  Dalton's  symbol  for,  3. 

Graham's  examination  of,  4. 

a  diatomic  gas,  10. 

velocity  of  molecules  of,  29, 

39. 

degrees  of  freedom  of  mol- 
ecules of,  48. 

ratio  of  specific  heats  of,  49, 

52,  205. 


INDEX. 


219 


Hydrogen,  expansion  of,  through 
porous  plug,  54. 

a ' '  more  than  perfect  gas, ' '  54. 

coefficient  of  viscosity  of,  63. 

free  path  of  molecules  of,  66. 

molecular  collisions  in,  67. 

critical  temperature  of,  84. 

aggregate  volume  of  molecules 

of,  135. 

maximum  density  of,  136. 

and  palladium,  137. 

size  of  molecules  of,  139. 

number  of  molecules  in  a  unit 

volume  of,  148. 

and  Dulong  and  Petit' s  law, 

157. 

as    the    primordial  element, 

159. 

vibration-periods  of,  186. 

Hyposulphite  of  soda,  surface  ten- 
sion of,  94. 

Indices  of  crystals,  rationality  of, 

126,  131. 

Inertia,  "explanation"  of,  153. 
Integration  of  equations,  197. 
Iodine,  atomic  weight  of,  162. 

Joule,  53,  54,  204. 

Judd,  Prof.  John  W.,  125. 

Kayser  and  Runge,  189. 

Kedzie,  Prof.  R.  C.,  185. 

Kelvin,  Lord,  19,  36,  45,  93,  140, 
143, 147,  174, 192,  194, 209.  (See 
also  Thomson,  Sir  William.) 

Kinetic  energy,  partition  of,  in  mol- 
ecules, 31,  35,  37,  122,  155,  188. 

of  translation,  relation 

of,  to  temperature,  41,  42,  45. 

Kinetic  theory  of  gases,  14. 

objections  to,  19,  49. 

fundamental  assump- 
tions of  original,  22. 


Kinetic  theory  of  gases,  fundamen- 
tal assumptions  of  generalized,  34. 

results  of ,  compared  with 

observation,  42. 

Knott,  62. 

Kundt  and  Warburg,  48,  62,  206. 

Le  Sage,  190. 

Leverrier,  163. 

Light,  elastic  solid  theory  of,  164. 

electro-magnetic    theory    of, 

168. 

corpuscular  theory  of,  169. 

Rankine's  theory  of,  173. 

Lindemann,  180. 

Liquid,  definition  of,  11. 

surface,  work  done  in  form- 
ing, 96. 

Liquids,  molecular  constitution  of, 
12. 

diagram  of  molecular  motion 

in,  13. 

molecular  theory  of,  74. 

surface  phenomena  of,  76. 

critical  points  of,  82. 

-  table  of,  84. 

contraction  and  compressibil- 
ity of,  84. 

surface  tension  of,  86. 

latent  heats  of  vaporization 

of,  96. 

"Liquide  glyce'rique,"   Plateau's, 

87,  206. 

Lithium,  atomic  weight  of,  162. 
Liveing,  Prof.,  127,  128. 
Lodge,  Dr.  Oliver,  171. 
Lucretius,  178. 

Magnitudes,  molecular,  133. 
Marsh  gas,  critical  temperature  of, 

84. 
Matter,  Dr.  Burton  on  the  origin  of, 

181. 


220 


INDEX. 


Maxwell,  31, 61, 62, 69, 94, 102, 106, 
120,  147,  155,  167,  169,  178,  185, 
193. 

Maxwell's  theorem,  24. 

Mendeleieff  and  Meyer,  periodic 
law  of,  161. 

Mercury,  degrees  of  freedom  of  mol- 
ecules of,  48. 

surface  tension  of,  94. 

latent  heat  of  vaporization  of, 

96. 

work  done  in  forming  more 

surface,  101. 

aggregate  volume  of  molecules 

of,  134. 

size  of  molecules  of,  145. 

and  Dulong  and  Pe tit's  law, 

157. 

ratio  of  specific  heats  of  vapor 

of,  48,  206. 

critical  temperature  of,  213. 

Meyer  (L.),  154,  159. 

Meyer  and    Mendeleieff,   periodic 

law  of,  161. 
Meyer  (0.),  61. 
Mixtures,  gaseous,  properties  of,  31, 

35,  40. 

Molecular  hypothesis,  1. 
requirements  of,  153. 

conditions,  existence  of  three 

principal,  12. 

constitution  of  solids,  12, 103, 

106,  127. 

of  liquids,  12,  74. 

of  gases,  14. 

and  stellar  distances,  14,  138. 

collisions,  15,  67. 

structure  of  crystals,  127. 

magnitudes,  133. 

illustrations  of,  149. 

attraction  in  gases,  53. 

range  of,  by  Quincke's 

wedge,  145. 


Molecular  attraction,  range  of,  by 
instability  of  liquid  films,  147. 

vortices,  Rankine's  hypothesis 

of,  173. 

Molecules,  similarity  of,  4. 

and  atoms,  distinction  of,  6. 

compound  nature  of,  9. 

of  gases,  distances  of,  14, 137. 

analogy  of  with  bees,  15. 

-  with  bullets,  17. 
-  elasticity  of,  15,  17,  172. 

coefficient  of  restitution  of, 

20. 

conception  of,  hi  original  ki- 
netic theory,  22. 

in  generalized  theory,  34. 

internal  vibratory  energy  of, 

19,  36,  182. 

degrees  of  freedom  of,  36,  37, 

156,  188. 

forces  between,  49,  53,  74. 

critical  velocity  of,  50,  82. 

arrangement  of,  in  solids,  12, 

103,  106. 

in  crystals,  127. 

"size"  of,  meaning  of  term, 

133. 

• aggregate  volume  of,  by  elec- 
trical method,  133. 
by  gas  equation,  134. 

mean  distances  of,  14,  137. 

size  of,  by  Clausius'  method, 

138. 

by  Lord  Kelvin's  elec- 
trical method,  139. 

by  camphor  movements, 

142. 

by  surface  tension  meth- 
od, 143. 

by  polarization  of  plat- 
inum, 147. 

by  polarization  of  silver 

by  iodine  vapor,  147. 


INDEX. 


221 


Molecules,  number  of,  in  a  unit 
volume  of  gas,  148. 

constitution  of,  53,  151. 

provisional  assumptions 

about,  171. 

Rankine's  hypothesis, 

173. 

Lord  Kelvin's  vortex 

theory,  174. 

Dr.  Burton's  strain-fig- 
ure theory,  180. 

Nascent  state,  9. 
Neptune,  discovery  of,  163. 
Newton,  Sir  Isaac,  169,  190,  196. 
Nitrogen,  Dalton's  symbol  for,  3. 

velocity  of  molecules  of,  40. 

degrees  of  freedom  of  mol- 
ecule of,  48. 

ratio  of  specific  heats  of,  49, 

52,  205. 

coefficient  of  viscosity  of,  63. 

free  path  of  molecules  of,  66. 

molecular  collisions  in,  67. 

critical  temperature  of,  84. 

aggregate  volume  of  molecules 

of,  135. 

maximum  density  of,  136. 

size  of  molecules  of,  139. 

number  of  molecules  per  unit 

volume,  148. 

and  Dulong  and  Petit' s  law, 

157. 

Nitrous  oxide,  critical  temperature 
of,  84. 

Oberbeck,  147. 

Observation,  results  of,  compared 

with  kinetic  theory,  42. 
Olive  oil,  surface  tension  of,  94. 

size  of  molecules  of,  143. 

Olszewski,  137. 

Osmotic  pressure,  117. 


Ostwald,  119. 

Oxygen,  Dalton's  symbol  for,  3. 

velocity  of  molecules  of,  40. 

degrees  of  freedom  of  mol- 
ecules of,  48. 

ratio  of  specific  heats  of,  49, 

52,  205. 

coefficient  of  viscosity  of,  63. 

free  path  of  molecules  of,  66. 

molecular  collisions  in,  67. 

critical  temperature  of,  84. 

aggregate  volume  of  molecules 

of,  135. 

maximum  density  of,  136. 

size  of  molecules  of,  139. 

number  of  molecules  per  unit 

volume,  148. 

and  Dulong  and  Petit's  law, 

157. 

Palladium  and  hydrogen,  137. 

Path,  free,  in  gases,  14,  15,  66. 

Periodic  law,  the,  161. 

Petroleum,  surface  tension  of,  94. 

Pfeffer,  117. 

Plateau,  145,  146,  206. 

Plug,  porous,  53. 

Poisson,  52. 

Potassium,  atomic  weight  of,  162. 

Pressure,  gaseous,  17,  38,  40. 

vapor,  79. 

Preston,  Thomas,  82,  136,  194. 
Prout's  hypothesis,  159. 

Quincke,  94,  145. 

Radiometer,  68. 
Rankine,  52,  80,  173,  203. 
Rationality  of  crystal  indices,  126, 

131. 

Rayleigh,  Lord,  95,  143,  168. 
Regnault,  46,  52,  54,  136. 
Reinold  and  Riicker,  147. 
Rigid  body,  freedom  of,  33. 


222 


INDEX. 


Sal  ammoniac,  dissociation  of,  9. 

Saturation,  121. 

Scandium,  163. 

Smoke  rings,  175. 

Soap  films,  phenomena  of,  87,  144, 

209. 

Sodium,  atomic  weight  of,  162. 
Sohncke,  127. 
Solids,  definition  of,  11. 

molecular  constitution  of,  12, 

103,  106,  127. 

theory  of,  102. 

stresses  in,  102. 

viscosity  of,  103. 

Solutions,  112. 

saturated,  121. 

supersaturated,  123. 

Specific  heats  of  gases,  ratio  of,  47, 

203. 

Spectroscope,  185. 
Stas,  4. 
Steam,  ratio  of  specific  heats  of,  52, 

206. 

Rankine's  formula  for  pres- 
sure of,  80. 

dissociation  of,  110. 

and  Dulong  and  Petit' s  law, 

157. 

Strain-figure  theory,  Dr.  Burton's, 

180. 

Strontium,  atomic  weight  of,  162. 
Sublimation,  108. 
Sulphur,  dissociation  of,  9. 

dioxide,  critical  temperature 

of,  84. 

Supersaturation,  123. 

Surface,  liquid,  phenomena  at  a,  76. 

work  done  in  forming, 

96. 

clean  water,  89. 

tension,  86. 

Surface  tension,  magnitude  of,  90. 
table  of,  94. 


Surface  tension  method  for  finding 

sizes  of  molecules,  143. 
Sutherland,  William,   55,   64,  84, 

135,  149. 

TABLES : 

Dalton's  atomic  weights,  3. 
Velocities  of  Spheres  (Maxwell's 

law),  26. 

Molecular  velocities  in  gases,  40. 
Comparison    of    kinetic    theory 

with  observation,  42. 
Corrections  to  reduce  thermom- 
eter readings  to  absolute  scale, 

46. 
Ratios  of  specific  heats  of  gases, 

49,  205. 
Coefficients  of  viscosity  of  gases, 

63. 
Holman's    experiments    on    the 

viscosity  of  C02,  65. 
Free  paths,  66. 

Molecular  collisions  in  gases,  67. 
Critical  temperatures,  84. 
Surface  tensions,  94. 
Latent    heats    of    vaporization, 

96. 

Work  done  in  forming  liquid  sur- 
face, 101. 
Maximum    densities    of    gases, 

136. 
Molecular   diameters    by    Clau- 

sius's  method,  139. 
by    surface    tension 

method,  145. 
Number  of  molecules  hi  a  unit 

volume  of  gas,  148r 
Dulong  and  Petit's  law,  155. 
Determination  of  (7,  157. 
Comparison  of    eka-boron    and 

scandium,  163. 
Balmer's  law,  186. 
Computation  of  m,  213. 


INDEX. 


223 


Tait,  P.  G.,  175. 
Target,  analogy  of,  17. 
Temperature,  41,  42,  43,  47. 

critical,  82. 

effect    of,   on    coefficient    of 

viscosity,  64. 

Tension  ;  —  see  Surface  tension. 
Thermal  phenomena  of  films,  209. 
Thermodynamics,  47. 
Thermometer  scales,  44. 
Thomson,  J.  J.,  183,  187. 
Thomson,  Sir  William,  52,  53,  54, 
103,  204. 

(See  also  Kelvin,  Lord.) 
Triads,  Dobereiner's,  162. 
Turpentine,  surface  tension  of,  94. 

Vacua,  high,  68. 
Vacuum  tubes,  70. 
Van  der  Waals,  56. 
Van't  Hoff,  119. 
Vapor  density,  77. 

pressure,  79. 

critical  temperatures  of,  82. 

Vaporization,  latent  heat  of,  95. 

(See  also  Evaporation.) 
Velocities,  molecular,  in  gases,  20, 

29,  39,  40. 
Maxwell's  law  of, 

24,  26. 
in  gaseous  mixtures,  35. 

critical,  of  molecules,  50,  82. 

Vibratory  energy  of  molecules,  19, 

36,  182. 

Viscosity  of  gases,  59,  64. 
coefficient  of,  60,  61. 


Viscosity  of  gases,  kinetic  explana- 
tion of,  63. 

of  solids,  103. 

Vortex  theory,  Lord  Kelvin's,  153, 

174,  194. 
Vortices,     molecular,     Rankine's 

theory  of,  173. 

Water,  Dalton's  symbol  for,  3. 

as  a  thermometric  fluid,  45. 

critical  temperature  of,  84. 

surface,  clean,  89. 

capillary  curve  of,  91. 

surface  tension  of,  94. 

latent  heat  of  vaporization  of, 

96. 

work  done  in  forming  more 

surface,  101. 

size  of  molecules  of,  145. 

vapor  and  Dulong  and  Petit's 

law,  157. 

ratio  of  specific  heats  of, 

52,  206. 
Watson,  50. 
Wedge,  Quincke's,  145. 
Weight,  conservation  of,  160. 
Wiener,  147. 
Williams,  124. 
Wires,  behavior  of,  under  torsion, 

102. 

Wolf,  95. 
Wroblevsky,  137. 


Zinc  and  copper,  size  of  molecules 


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